{"id":887,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=887"},"modified":"2015-11-12T18:37:58","modified_gmt":"2015-11-12T18:37:58","slug":"use-a-graph-to-locate-the-absolute-maximum-and-absolute-minimum","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/use-a-graph-to-locate-the-absolute-maximum-and-absolute-minimum\/","title":{"raw":"Use a graph to locate the absolute maximum and absolute minimum","rendered":"Use a graph to locate the absolute maximum and absolute minimum"},"content":{"raw":"

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the absolute maximum <\/strong>and absolute minimum<\/strong>, respectively.<\/p>\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.\n<\/span><\/span>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 10<\/b>[\/caption]\n

Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n\n

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A General Note: Absolute Maxima and Minima<\/h3>\n

The absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n

The absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n\n<\/div>\n

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Example 10: Finding Absolute Maxima and Minima from a Graph<\/h3>\nFor the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.\n<\/span><\/span>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 11<\/b>[\/caption]\n\n<\/div>\n
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Solution<\/h3>\n

Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\n

The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n

","rendered":"

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the absolute maximum <\/strong>and absolute minimum<\/strong>, respectively.<\/p>\n

To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.
\n<\/span><\/span><\/p>\n

\"Graph<\/p>\n

Figure 10<\/b><\/p>\n<\/div>\n

Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n

\n

A General Note: Absolute Maxima and Minima<\/h3>\n

The absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n

The absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n

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Example 10: Finding Absolute Maxima and Minima from a Graph<\/h3>\n

For the function [latex]f[\/latex] shown in Figure 11, find all absolute maxima and minima.
\n<\/span><\/span><\/p>\n

\"Graph<\/p>\n

Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n

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Solution<\/h3>\n

Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\n

The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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