{"id":485,"date":"2021-02-04T15:30:47","date_gmt":"2021-02-04T15:30:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=485"},"modified":"2021-04-09T01:47:51","modified_gmt":"2021-04-09T01:47:51","slug":"problem-set-maxima-and-minima","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-maxima-and-minima\/","title":{"raw":"Problem Set: Maxima and Minima","rendered":"Problem Set: Maxima and Minima"},"content":{"raw":"
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1.<\/strong> In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation [latex]y=ax^2+bx+c[\/latex], which was [latex]m=-\\frac{b}{2a}[\/latex]. Prove this formula using calculus.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.<\/strong> If you are finding an absolute minimum over an interval [latex][a,b][\/latex], why do you need to check the endpoints? Draw a graph that supports your hypothesis.<\/p>\r\n[reveal-answer q=\"fs-id1165042199477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042199477\"]\r\n

Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.<\/strong> If you are examining a function over an interval [latex](a,b)[\/latex], for [latex]a[\/latex] and [latex]b[\/latex] finite, is it possible not to have an absolute maximum or absolute minimum?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.<\/strong> When you are checking for critical points, explain why you also need to determine points where [latex]f^{\\prime}(x)[\/latex] is undefined. Draw a graph to support your explanation.<\/p>\r\n[reveal-answer q=\"fs-id1165042278285\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042278285\"]\r\n

Answers will vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.<\/strong> Can you have a finite absolute maximum for [latex]y=ax^2+bx+c[\/latex] over [latex](\u2212\\infty ,\\infty )[\/latex]? Explain why or why not using graphical arguments.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.<\/strong> Can you have a finite absolute maximum for [latex]y=ax^3+bx^2+cx+d[\/latex] over [latex](\u2212\\infty ,\\infty )[\/latex] assuming [latex]a[\/latex] is non-zero? Explain why or why not using graphical arguments.<\/p>\r\n[reveal-answer q=\"fs-id1165042278437\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042278437\"]\r\n

No; answers will vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.<\/strong> Let [latex]m[\/latex] be the number of local minima and [latex]M[\/latex] be the number of local maxima. Can you create a function where [latex]M>m+2[\/latex]? Draw a graph to support your explanation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.<\/strong> Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.<\/p>\r\n[reveal-answer q=\"fs-id1165042065918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042065918\"]\r\n

Since the absolute maximum is the function (output) value rather than the [latex]x[\/latex] value, the answer is no; answers will vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.<\/strong> Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [T]<\/strong> Graph the function [latex]y=e^{ax}[\/latex]. For which values of [latex]a[\/latex], on any infinite domain, will you have an absolute minimum and absolute maximum?<\/p>\r\n[reveal-answer q=\"fs-id1165042065985\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042065985\"]\r\n

When [latex]a=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (11-14), determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function.<\/p>\r\n\r\n

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11.<\/strong>\r\n\"The<\/span><\/div>\r\n<\/div>\r\n
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12.<\/strong>\r\n\"The\r\n[reveal-answer q=\"fs-id1165042066053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042066053\"]Absolute minimum at 3; Absolute maximum at \u22122.2; local minima at \u22122, 1; local maxima at \u22121, 2[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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13.<\/strong>\r\n\"The<\/span><\/div>\r\n<\/div>\r\n
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14.<\/strong>\r\n\"The\r\n[reveal-answer q=\"fs-id1165040665497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040665497\"]Absolute minima at \u22122, 2; absolute maxima at \u22122.5, 2.5; local minimum at 0; local maxima at \u22121, 1[\/hidden-answer]<\/div>\r\n<\/div>\r\n

For the following problems (15-18), draw graphs of [latex]f(x)[\/latex], which is continuous, over the interval [latex][-4,4][\/latex] with the following properties:<\/p>\r\n\r\n

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15.<\/strong> Absolute maximum at [latex]x=2[\/latex] and absolute minima at [latex]x=\\pm 3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.<\/strong> Absolute minimum at [latex]x=1[\/latex] and absolute maximum at [latex]x=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165040665606\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040665606\"]\r\n

Answers may vary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.<\/strong> Absolute maximum at [latex]x=4[\/latex], absolute minimum at [latex]x=-1[\/latex], local maximum at [latex]x=-2[\/latex], and a critical point that is not a maximum or minimum at [latex]x=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.<\/strong> Absolute maxima at [latex]x=2[\/latex] and [latex]x=-3[\/latex], local minimum at [latex]x=1[\/latex], and absolute minimum at [latex]x=4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042110003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042110003\"]\r\n

Answers may vary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (19-28), find the critical points in the domains of the following functions.<\/p>\r\n\r\n

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19.<\/strong> [latex]y=4x^3-3x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.<\/strong> [latex]y=4\\sqrt{x}-x^2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042110089\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042110089\"]\r\n

[latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.<\/strong> [latex]y=\\frac{1}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.<\/strong> [latex]y=\\ln (x-2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042110167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042110167\"]\r\n

None<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.<\/strong> [latex]y= \\tan (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.<\/strong> [latex]y=\\sqrt{4-x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165040750576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040750576\"]\r\n

[latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.<\/strong> [latex]y=x^{3\/2}-3x^{5\/2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.<\/strong> [latex]y=\\frac{x^2-1}{x^2+2x-3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165040750701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040750701\"]\r\n

None<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.<\/strong> [latex]y= \\sin^2 (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.<\/strong> [latex]y=x+\\frac{1}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165041864898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165041864898\"]\r\n

[latex]x=-1,1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (29-39), find the local and\/or absolute maxima for the functions over the specified domain.<\/p>\r\n\r\n

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29.<\/strong> [latex]f(x)=x^2+3[\/latex] over [latex][-1,4][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.<\/strong> [latex]y=x^2+\\frac{2}{x}[\/latex] over [latex][1,4][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165041865087\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165041865087\"]\r\n

Absolute maximum: [latex]x=4[\/latex], [latex]y=\\frac{33}{2}[\/latex]; absolute minimum: [latex]x=1[\/latex], [latex]y=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.<\/strong> [latex]y=(x-x^2)^2[\/latex] over [latex][-1,1][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.<\/strong> [latex]y=\\frac{1}{(x-x^2)}[\/latex] over [latex](0,1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042051184\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042051184\"]\r\n

Absolute minimum: [latex]x=\\frac{1}{2}[\/latex], [latex]y=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.<\/strong> [latex]y=\\sqrt{9-x}[\/latex] over [latex][1,9][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.<\/strong> [latex]y=x+ \\sin (x)[\/latex] over [latex][0,2\\pi][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042051358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042051358\"]\r\n

Absolute maximum: [latex]x=2\\pi [\/latex], [latex]y=2\\pi [\/latex]; absolute minimum: [latex]x=0[\/latex], [latex]y=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35.<\/strong> [latex]y=\\frac{x}{1+x}[\/latex] over [latex][0,100][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.<\/strong> [latex]y=|x+1|+|x-1|[\/latex] over [latex][-3,2][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042062131\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042062131\"]\r\n

Absolute maximum: [latex]x=-3[\/latex]; absolute minimum: [latex]-1\\le x\\le 1[\/latex], [latex]y=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.<\/strong> [latex]y=\\sqrt{x}-\\sqrt{x^3}[\/latex] over [latex][0,4][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.<\/strong> [latex]y= \\sin x+ \\cos x[\/latex] over [latex][0,2\\pi][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042288713\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042288713\"]\r\n

Absolute maximum: [latex]x=\\frac{\\pi}{4}[\/latex], [latex]y=\\sqrt{2}[\/latex]; absolute minimum: [latex]x=\\frac{5\\pi}{4}[\/latex], [latex]y=\u2212\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.<\/strong> [latex]y=4 \\sin \\theta -3 \\cos \\theta [\/latex] over [latex][0,2\\pi][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (40-45), find the local and absolute minima and maxima for the functions over [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\r\n\r\n

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40.<\/strong> [latex]y=x^2+4x+5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165040756213\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040756213\"]\r\n

Absolute minimum: [latex]x=-2[\/latex], [latex]y=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41.<\/strong> [latex]y=x^3-12x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42.<\/strong> [latex]y=3x^4+8x^3-18x^2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042266619\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042266619\"]\r\n

Absolute minimum: [latex]x=-3[\/latex], [latex]y=-135[\/latex]; local maximum: [latex]x=0[\/latex], [latex]y=0[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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43.<\/strong> [latex]y=x^3(1-x)^6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.<\/strong> [latex]y=\\dfrac{x^2+x+6}{x-1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165040675956\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040675956\"]\r\n

Local maximum: [latex]x=1-2\\sqrt{2}[\/latex], [latex]y=3-4\\sqrt{2}[\/latex]; local minimum: [latex]x=1+2\\sqrt{2}[\/latex], [latex]y=3+4\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.<\/strong> [latex]y=\\dfrac{x^2-1}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.<\/p>\r\n\r\n

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46. [T]\u00a0<\/strong>[latex]y=3x\\sqrt{1-x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042295764\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042295764\"]\r\n

Absolute maximum: [latex]x=\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=\\frac{3}{2}[\/latex]; absolute minimum: [latex]x=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=-\\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [T]\u00a0<\/strong>[latex]y=x+ \\sin (x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. [T]\u00a0<\/strong>[latex]y=12x^5+45x^4+20x^3-90x^2-120x+3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042295945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042295945\"]\r\n

Local maximum: [latex]x=-2[\/latex], [latex]y=59[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-130[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [T]\u00a0<\/strong>[latex]y=\\dfrac{x^3+6x^2-x-30}{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50. [T]\u00a0<\/strong>[latex]y=\\dfrac{\\sqrt{4-x^2}}{\\sqrt{4+x^2}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042231095\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042231095\"]\r\n

Absolute maximum: [latex]x=0[\/latex], [latex]y=1[\/latex]; absolute minimum: [latex]x=-2,2[\/latex], [latex]y=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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51.<\/strong> A company that produces cell phones has a cost function of [latex]C=x^2-1200x+36,400[\/latex], where [latex]C[\/latex] is cost in dollars and [latex]x[\/latex] is number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes this cost function?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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52.<\/strong> A ball is thrown into the air and its position is given by [latex]h(t)=-4.9t^2+60t+5[\/latex] m. Find the height at which the ball stops ascending. How long after it is thrown does this happen?<\/p>\r\n[reveal-answer q=\"fs-id1165042237916\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042237916\"]\r\n

[latex]h=\\frac{9245}{49}[\/latex] m, [latex]t=\\frac{300}{49}[\/latex] sec.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (53-54), consider the production of gold during the California gold rush (1848\u20131888). The production of gold can be modeled by [latex]G(t)=\\frac{(25t)}{(t^2+16)}[\/latex], where [latex]t[\/latex] is the number of years since the rush began [latex](0\\le t\\le 40)[\/latex] and [latex]G[\/latex] is ounces of gold produced (in millions). A summary of the data is shown in the following figure.<\/p>\r\n\"The<\/span>\r\n

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53.<\/strong> Find when the maximum (local and global) gold production occurred, and the amount of gold produced during that maximum.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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54.<\/strong> Find when the minimum (local and global) gold production occurred. What was the amount of gold produced during this minimum?<\/p>\r\n[reveal-answer q=\"fs-id1165042304259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042304259\"]\r\n

The global minimum was in 1848, when no gold was produced.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the critical points, maxima, and minima for the following piecewise functions (55-56).<\/p>\r\n\r\n

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55.<\/strong> [latex]y=\\begin{cases} x^2-4x, & 0\\le x\\le 1\\\\ x^2-4, & 1<x\\le 2 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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56.<\/strong> [latex]y=\\begin{cases} x^2+1, & x\\le 1\\\\ x^2-4x+5, & x>1 \\end{cases}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042304474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042304474\"]\r\n

Absolute minima: [latex]x=0[\/latex], [latex]x=2[\/latex], [latex]y=1[\/latex]; local maximum at [latex]x=1[\/latex], [latex]y=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (57-58), find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions.<\/p>\r\n\r\n

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57.<\/strong> [latex]y=ax^2+bx+c[\/latex], given that [latex]a>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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58.<\/strong> [latex]y=(x-1)^a[\/latex], given that [latex]a>1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042059022\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042059022\"]\r\n

No maxima\/minima if [latex]a[\/latex] is odd, minimum at [latex]x=1[\/latex] if [latex]a[\/latex] is even<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

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1.<\/strong> In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation [latex]y=ax^2+bx+c[\/latex], which was [latex]m=-\\frac{b}{2a}[\/latex]. Prove this formula using calculus.<\/p>\n<\/div>\n<\/div>\n

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2.<\/strong> If you are finding an absolute minimum over an interval [latex][a,b][\/latex], why do you need to check the endpoints? Draw a graph that supports your hypothesis.<\/p>\n

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Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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3.<\/strong> If you are examining a function over an interval [latex](a,b)[\/latex], for [latex]a[\/latex] and [latex]b[\/latex] finite, is it possible not to have an absolute maximum or absolute minimum?<\/p>\n<\/div>\n<\/div>\n

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4.<\/strong> When you are checking for critical points, explain why you also need to determine points where [latex]f^{\\prime}(x)[\/latex] is undefined. Draw a graph to support your explanation.<\/p>\n

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Answers will vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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5.<\/strong> Can you have a finite absolute maximum for [latex]y=ax^2+bx+c[\/latex] over [latex](\u2212\\infty ,\\infty )[\/latex]? Explain why or why not using graphical arguments.<\/p>\n<\/div>\n<\/div>\n

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6.<\/strong> Can you have a finite absolute maximum for [latex]y=ax^3+bx^2+cx+d[\/latex] over [latex](\u2212\\infty ,\\infty )[\/latex] assuming [latex]a[\/latex] is non-zero? Explain why or why not using graphical arguments.<\/p>\n

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No; answers will vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7.<\/strong> Let [latex]m[\/latex] be the number of local minima and [latex]M[\/latex] be the number of local maxima. Can you create a function where [latex]M>m+2[\/latex]? Draw a graph to support your explanation.<\/p>\n<\/div>\n<\/div>\n

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8.<\/strong> Is it possible to have more than one absolute maximum? Use a graphical argument to prove your hypothesis.<\/p>\n

Show Solution<\/span><\/p>\n
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Since the absolute maximum is the function (output) value rather than the [latex]x[\/latex] value, the answer is no; answers will vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.<\/strong> Is it possible to have no absolute minimum or maximum for a function? If so, construct such a function. If not, explain why this is not possible.<\/p>\n<\/div>\n<\/div>\n

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10. [T]<\/strong> Graph the function [latex]y=e^{ax}[\/latex]. For which values of [latex]a[\/latex], on any infinite domain, will you have an absolute minimum and absolute maximum?<\/p>\n

Show Solution<\/span><\/p>\n
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When [latex]a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (11-14), determine where the local and absolute maxima and minima occur on the graph given. Assume the graph represents the entirety of each function.<\/p>\n

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11.<\/strong>
\n\"The<\/span><\/div>\n<\/div>\n
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12.<\/strong>
\n\"The<\/p>\n
Show Solution<\/span><\/p>\n
Absolute minimum at 3; Absolute maximum at \u22122.2; local minima at \u22122, 1; local maxima at \u22121, 2<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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13.<\/strong>
\n\"The<\/span><\/div>\n<\/div>\n
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14.<\/strong>
\n\"The<\/p>\n
Show Solution<\/span><\/p>\n
Absolute minima at \u22122, 2; absolute maxima at \u22122.5, 2.5; local minimum at 0; local maxima at \u22121, 1<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following problems (15-18), draw graphs of [latex]f(x)[\/latex], which is continuous, over the interval [latex][-4,4][\/latex] with the following properties:<\/p>\n

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15.<\/strong> Absolute maximum at [latex]x=2[\/latex] and absolute minima at [latex]x=\\pm 3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.<\/strong> Absolute minimum at [latex]x=1[\/latex] and absolute maximum at [latex]x=2[\/latex]<\/p>\n

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Answers may vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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17.<\/strong> Absolute maximum at [latex]x=4[\/latex], absolute minimum at [latex]x=-1[\/latex], local maximum at [latex]x=-2[\/latex], and a critical point that is not a maximum or minimum at [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.<\/strong> Absolute maxima at [latex]x=2[\/latex] and [latex]x=-3[\/latex], local minimum at [latex]x=1[\/latex], and absolute minimum at [latex]x=4[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers may vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (19-28), find the critical points in the domains of the following functions.<\/p>\n

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19.<\/strong> [latex]y=4x^3-3x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20.<\/strong> [latex]y=4\\sqrt{x}-x^2[\/latex]<\/p>\n

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[latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21.<\/strong> [latex]y=\\frac{1}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.<\/strong> [latex]y=\\ln (x-2)[\/latex]<\/p>\n

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None<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.<\/strong> [latex]y= \\tan (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24.<\/strong> [latex]y=\\sqrt{4-x^2}[\/latex]<\/p>\n

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[latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.<\/strong> [latex]y=x^{3\/2}-3x^{5\/2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.<\/strong> [latex]y=\\frac{x^2-1}{x^2+2x-3}[\/latex]<\/p>\n

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None<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.<\/strong> [latex]y= \\sin^2 (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.<\/strong> [latex]y=x+\\frac{1}{x}[\/latex]<\/p>\n

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[latex]x=-1,1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (29-39), find the local and\/or absolute maxima for the functions over the specified domain.<\/p>\n

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29.<\/strong> [latex]f(x)=x^2+3[\/latex] over [latex][-1,4][\/latex]<\/p>\n<\/div>\n<\/div>\n

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30.<\/strong> [latex]y=x^2+\\frac{2}{x}[\/latex] over [latex][1,4][\/latex]<\/p>\n

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Absolute maximum: [latex]x=4[\/latex], [latex]y=\\frac{33}{2}[\/latex]; absolute minimum: [latex]x=1[\/latex], [latex]y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.<\/strong> [latex]y=(x-x^2)^2[\/latex] over [latex][-1,1][\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.<\/strong> [latex]y=\\frac{1}{(x-x^2)}[\/latex] over [latex](0,1)[\/latex]<\/p>\n

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Absolute minimum: [latex]x=\\frac{1}{2}[\/latex], [latex]y=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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33.<\/strong> [latex]y=\\sqrt{9-x}[\/latex] over [latex][1,9][\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.<\/strong> [latex]y=x+ \\sin (x)[\/latex] over [latex][0,2\\pi][\/latex]<\/p>\n

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Absolute maximum: [latex]x=2\\pi[\/latex], [latex]y=2\\pi[\/latex]; absolute minimum: [latex]x=0[\/latex], [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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35.<\/strong> [latex]y=\\frac{x}{1+x}[\/latex] over [latex][0,100][\/latex]<\/p>\n<\/div>\n<\/div>\n

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36.<\/strong> [latex]y=|x+1|+|x-1|[\/latex] over [latex][-3,2][\/latex]<\/p>\n

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Absolute maximum: [latex]x=-3[\/latex]; absolute minimum: [latex]-1\\le x\\le 1[\/latex], [latex]y=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37.<\/strong> [latex]y=\\sqrt{x}-\\sqrt{x^3}[\/latex] over [latex][0,4][\/latex]<\/p>\n<\/div>\n<\/div>\n

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38.<\/strong> [latex]y= \\sin x+ \\cos x[\/latex] over [latex][0,2\\pi][\/latex]<\/p>\n

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Absolute maximum: [latex]x=\\frac{\\pi}{4}[\/latex], [latex]y=\\sqrt{2}[\/latex]; absolute minimum: [latex]x=\\frac{5\\pi}{4}[\/latex], [latex]y=\u2212\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39.<\/strong> [latex]y=4 \\sin \\theta -3 \\cos \\theta[\/latex] over [latex][0,2\\pi][\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (40-45), find the local and absolute minima and maxima for the functions over [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\n

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40.<\/strong> [latex]y=x^2+4x+5[\/latex]<\/p>\n

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Absolute minimum: [latex]x=-2[\/latex], [latex]y=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41.<\/strong> [latex]y=x^3-12x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42.<\/strong> [latex]y=3x^4+8x^3-18x^2[\/latex]<\/p>\n

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Absolute minimum: [latex]x=-3[\/latex], [latex]y=-135[\/latex]; local maximum: [latex]x=0[\/latex], [latex]y=0[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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43.<\/strong> [latex]y=x^3(1-x)^6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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44.<\/strong> [latex]y=\\dfrac{x^2+x+6}{x-1}[\/latex]<\/p>\n

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Local maximum: [latex]x=1-2\\sqrt{2}[\/latex], [latex]y=3-4\\sqrt{2}[\/latex]; local minimum: [latex]x=1+2\\sqrt{2}[\/latex], [latex]y=3+4\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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45.<\/strong> [latex]y=\\dfrac{x^2-1}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following functions, use a calculator to graph the function and to estimate the absolute and local maxima and minima. Then, solve for them explicitly.<\/p>\n

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46. [T]\u00a0<\/strong>[latex]y=3x\\sqrt{1-x^2}[\/latex]<\/p>\n

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Absolute maximum: [latex]x=\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=\\frac{3}{2}[\/latex]; absolute minimum: [latex]x=-\\frac{\\sqrt{2}}{2}[\/latex], [latex]y=-\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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47. [T]\u00a0<\/strong>[latex]y=x+ \\sin (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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48. [T]\u00a0<\/strong>[latex]y=12x^5+45x^4+20x^3-90x^2-120x+3[\/latex]<\/p>\n

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Local maximum: [latex]x=-2[\/latex], [latex]y=59[\/latex]; local minimum: [latex]x=1[\/latex], [latex]y=-130[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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49. [T]\u00a0<\/strong>[latex]y=\\dfrac{x^3+6x^2-x-30}{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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50. [T]\u00a0<\/strong>[latex]y=\\dfrac{\\sqrt{4-x^2}}{\\sqrt{4+x^2}}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Absolute maximum: [latex]x=0[\/latex], [latex]y=1[\/latex]; absolute minimum: [latex]x=-2,2[\/latex], [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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51.<\/strong> A company that produces cell phones has a cost function of [latex]C=x^2-1200x+36,400[\/latex], where [latex]C[\/latex] is cost in dollars and [latex]x[\/latex] is number of cell phones produced (in thousands). How many units of cell phone (in thousands) minimizes this cost function?<\/p>\n<\/div>\n<\/div>\n

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52.<\/strong> A ball is thrown into the air and its position is given by [latex]h(t)=-4.9t^2+60t+5[\/latex] m. Find the height at which the ball stops ascending. How long after it is thrown does this happen?<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]h=\\frac{9245}{49}[\/latex] m, [latex]t=\\frac{300}{49}[\/latex] sec.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (53-54), consider the production of gold during the California gold rush (1848\u20131888). The production of gold can be modeled by [latex]G(t)=\\frac{(25t)}{(t^2+16)}[\/latex], where [latex]t[\/latex] is the number of years since the rush began [latex](0\\le t\\le 40)[\/latex] and [latex]G[\/latex] is ounces of gold produced (in millions). A summary of the data is shown in the following figure.<\/p>\n

\"The<\/span><\/p>\n

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53.<\/strong> Find when the maximum (local and global) gold production occurred, and the amount of gold produced during that maximum.<\/p>\n<\/div>\n<\/div>\n

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54.<\/strong> Find when the minimum (local and global) gold production occurred. What was the amount of gold produced during this minimum?<\/p>\n

Show Solution<\/span><\/p>\n
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The global minimum was in 1848, when no gold was produced.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Find the critical points, maxima, and minima for the following piecewise functions (55-56).<\/p>\n

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55.<\/strong> [latex]y=\\begin{cases} x^2-4x, & 0\\le x\\le 1\\\\ x^2-4, & 1\n<\/div>\n<\/div>\n

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56.<\/strong> [latex]y=\\begin{cases} x^2+1, & x\\le 1\\\\ x^2-4x+5, & x>1 \\end{cases}[\/latex]<\/p>\n

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Absolute minima: [latex]x=0[\/latex], [latex]x=2[\/latex], [latex]y=1[\/latex]; local maximum at [latex]x=1[\/latex], [latex]y=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (57-58), find the critical points of the following generic functions. Are they maxima, minima, or neither? State the necessary conditions.<\/p>\n

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57.<\/strong> [latex]y=ax^2+bx+c[\/latex], given that [latex]a>0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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58.<\/strong> [latex]y=(x-1)^a[\/latex], given that [latex]a>1[\/latex]<\/p>\n

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No maxima\/minima if [latex]a[\/latex] is odd, minimum at [latex]x=1[\/latex] if [latex]a[\/latex] is even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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