The First Derivative Test and Concavity

Learning Outcomes

  • Explain how the sign of the first derivative affects the shape of a function’s graph
  • State the first derivative test for critical points
  • Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph
  • Explain the concavity test for a function over an open interval

The First Derivative Test

Corollary 3 of the Mean Value Theorem showed that if the derivative of a function is positive over an interval [latex]I[/latex] then the function is increasing over [latex]I[/latex]. On the other hand, if the derivative of the function is negative over an interval [latex]I[/latex], then the function is decreasing over [latex]I[/latex] as shown in the following figure.

This figure is broken into four figures labeled a, b, c, and d. Figure a shows a function increasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f’ > 0. In other words, f is increasing. Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f’ > 0. In other words, f is increasing. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f’ < 0. In other words, f is decreasing. Figure d shows a function decreasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and it is noted that at both f’ < 0. In other words, f is decreasing.

Figure 1. Both functions are increasing over the interval [latex](a,b)[/latex]. At each point [latex]x[/latex], the derivative [latex]f^{\prime}(x)>0[/latex]. Both functions are decreasing over the interval [latex](a,b)[/latex]. At each point [latex]x[/latex], the derivative [latex]f^{\prime}(x)<0[/latex].

A continuous function [latex]f[/latex] has a local maximum at point [latex]c[/latex] if and only if [latex]f[/latex] switches from increasing to decreasing at point [latex]c[/latex]. Similarly, [latex]f[/latex] has a local minimum at [latex]c[/latex] if and only if [latex]f[/latex] switches from decreasing to increasing at [latex]c[/latex]. If [latex]f[/latex] is a continuous function over an interval [latex]I[/latex] containing [latex]c[/latex] and differentiable over [latex]I[/latex], except possibly at [latex]c[/latex], the only way [latex]f[/latex] can switch from increasing to decreasing (or vice versa) at point [latex]c[/latex] is if [latex]{f}^{\prime }[/latex] changes sign as [latex]x[/latex] increases through [latex]c.[/latex] If [latex]f[/latex] is differentiable at [latex]c,[/latex] the only way that [latex]{f}^{\prime }.[/latex] can change sign as [latex]x[/latex] increases through [latex]c[/latex] is if [latex]f^{\prime}(c)=0[/latex]. Therefore, for a function [latex]f[/latex] that is continuous over an interval [latex]I[/latex] containing [latex]c[/latex] and differentiable over [latex]I[/latex], except possibly at [latex]c[/latex], the only way [latex]f[/latex] can switch from increasing to decreasing (or vice versa) is if [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. Consequently, to locate local extrema for a function [latex]f[/latex], we look for points [latex]c[/latex] in the domain of [latex]f[/latex] such that [latex]f^{\prime}(c)=0[/latex] or [latex]f^{\prime}(c)[/latex] is undefined. Recall that such points are called critical points of [latex]f[/latex].

Note that [latex]f[/latex] need not have a local extrema at a critical point. The critical points are candidates for local extrema only. In Figure 2, we show that if a continuous function [latex]f[/latex] has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. We show that if [latex]f[/latex] has a local extremum at a critical point, then the sign of [latex]f^{\prime}[/latex] switches as [latex]x[/latex] increases through that point.

A function f(x) is graphed. It starts in the second quadrant and increases to x = a, which is too sharp and hence f’(a) is undefined. In this section f’ > 0. Then, f decreases from x = a to x = b (so f’ < 0 here), before increasing at x = b. It is noted that f’(b) = 0. While increasing from x = b to x = c, f’ > 0. The function has an inversion point at c, and it is marked f’(c) = 0. The function increases some more to d (so f’ > 0), which is the global maximum. It is marked that f’(d) = 0. Then the function decreases and it is marked that f’ > 0.

Figure 2. The function [latex]f[/latex] has four critical points: [latex]a,b,c[/latex], and [latex]d[/latex]. The function [latex]f[/latex] has local maxima at [latex]a[/latex] and [latex]d[/latex], and a local minimum at [latex]b[/latex]. The function [latex]f[/latex] does not have a local extremum at [latex]c[/latex]. The sign of [latex]f^{\prime}[/latex] changes at all local extrema.

Using Figure 2, we summarize the main results regarding local extrema.

  • If a continuous function [latex]f[/latex] has a local extremum, it must occur at a critical point [latex]c[/latex].
  • The function has a local extremum at the critical point [latex]c[/latex] if and only if the derivative [latex]f^{\prime}[/latex] switches sign as [latex]x[/latex] increases through [latex]c[/latex].
  • Therefore, to test whether a function has a local extremum at a critical point [latex]c[/latex], we must determine the sign of [latex]f^{\prime}(x)[/latex] to the left and right of [latex]c[/latex].

This result is known as the first derivative test.

First Derivative Test

Suppose that [latex]f[/latex] is a continuous function over an interval [latex]I[/latex] containing a critical point [latex]c[/latex]. If [latex]f[/latex] is differentiable over [latex]I[/latex], except possibly at point [latex]c[/latex], then [latex]f(c)[/latex] satisfies one of the following descriptions:

  1. If [latex]f^{\prime}[/latex] changes sign from positive when [latex]x<c[/latex] to negative when [latex]x>c[/latex], then [latex]f(c)[/latex] is a local maximum of [latex]f[/latex].
  2. If [latex]f^{\prime}[/latex] changes sign from negative when [latex]x<c[/latex] to positive when [latex]x>c[/latex], then [latex]f(c)[/latex] is a local minimum of [latex]f[/latex].
  3. If [latex]f^{\prime}[/latex] has the same sign for [latex]x<c[/latex] and [latex]x>c[/latex], then [latex]f(c)[/latex] is neither a local maximum nor a local minimum of [latex]f[/latex].

We can summarize the first derivative test as a strategy for locating local extrema.

Problem-Solving Strategy: Using the First Derivative Test

Consider a function [latex]f[/latex] that is continuous over an interval [latex]I[/latex].

  1. Find all critical points of [latex]f[/latex] and divide the interval [latex]I[/latex] into smaller intervals using the critical points as endpoints.
  2. Analyze the sign of [latex]f^{\prime}[/latex] in each of the subintervals. If [latex]f^{\prime}[/latex] is continuous over a given subinterval (which is typically the case), then the sign of [latex]f^{\prime}[/latex] in that subinterval does not change and, therefore, can be determined by choosing an arbitrary test point [latex]x[/latex] in that subinterval and by evaluating the sign of [latex]f^{\prime}[/latex] at that test point. Use the sign analysis to determine whether [latex]f[/latex] is increasing or decreasing over that interval.
  3. Use the first derivative test and the results of step 2 to determine whether [latex]f[/latex] has a local maximum, a local minimum, or neither at each of the critical points.

Recall from Chapter 4.3 that when talking about local extrema, the value of the extremum is the y value and the location of the extremum is the x value.

Now let’s look at how to use this strategy to locate all local extrema for particular functions.

Example: Using the First Derivative Test to Find Local Extrema

Use the first derivative test to find the location of all local extrema for [latex]f(x)=x^3-3x^2-9x-1[/latex]. Use a graphing utility to confirm your results.

Watch the following video to see the worked solution to Example: Using the First Derivative Test to Find Local Extrema.

Try It

Use the first derivative test to locate all local extrema for [latex]f(x)=−x^3+\frac{3}{2}x^2+18x[/latex].

Example: Using the First Derivative Test

Use the first derivative test to find the location of all local extrema for [latex]f(x)=5x^{\frac{1}{3}}-x^{\frac{5}{3}}[/latex]. Use a graphing utility to confirm your results.

Try It

Use the first derivative test to find all local extrema for [latex]f(x)=\sqrt[3]{x-1}[/latex].

Try It

Try It

Concavity and Points of Inflection

We now know how to determine where a function is increasing or decreasing. However, there is another issue to consider regarding the shape of the graph of a function. If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function.

Figure 5(a) shows a function [latex]f[/latex] with a graph that curves upward. As [latex]x[/latex] increases, the slope of the tangent line increases. Thus, since the derivative increases as [latex]x[/latex] increases, [latex]f^{\prime}[/latex] is an increasing function. We say this function [latex]f[/latex] is concave up. Figure 5(b) shows a function [latex]f[/latex] that curves downward. As [latex]x[/latex] increases, the slope of the tangent line decreases. Since the derivative decreases as [latex]x[/latex] increases, [latex]f^{\prime}[/latex] is a decreasing function. We say this function [latex]f[/latex] is concave down.

Definition

Let [latex]f[/latex] be a function that is differentiable over an open interval [latex]I[/latex]. If [latex]f^{\prime}[/latex] is increasing over [latex]I[/latex], we say [latex]f[/latex] is concave up over [latex]I[/latex]. If [latex]f^{\prime}[/latex] is decreasing over [latex]I[/latex], we say [latex]f[/latex] is concave down over [latex]I[/latex].

This figure is broken into four figures labeled a, b, c, and d. Figure a shows a function increasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing more. It is noted that f’ is increasing and f is concave up. Figure b shows a function increasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are increasing, but the one taken further to the right is increasing less. It is noted that f’ is decreasing and f is concave down. Figure c shows a function decreasing concavely from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing less. It is noted that f’ is increasing and f is concave up. Figure d shows a function decreasing convexly from (a, f(a)) to (b, f(b)). At two points the derivative is taken and both are decreasing, but the one taken further to the right is decreasing more. It is noted that f’ is decreasing and f is concave down.

Figure 5. (a), (c) Since [latex]f^{\prime}[/latex] is increasing over the interval [latex](a,b)[/latex], we say [latex]f[/latex] is concave up over [latex](a,b)[/latex]. (b), (d) Since [latex]f^{\prime}[/latex] is decreasing over the interval [latex](a,b)[/latex], we say [latex]f[/latex] is concave down over [latex](a,b)[/latex].

In general, without having the graph of a function [latex]f[/latex], how can we determine its concavity? By definition, a function [latex]f[/latex] is concave up if [latex]f^{\prime}[/latex] is increasing. From Corollary 3, we know that if [latex]f^{\prime}[/latex] is a differentiable function, then [latex]f^{\prime}[/latex] is increasing if its derivative [latex]f^{\prime \prime}(x)>0[/latex]. Therefore, a function [latex]f[/latex] that is twice differentiable is concave up when [latex]f^{\prime \prime}(x)>0[/latex]. Similarly, a function [latex]f[/latex] is concave down if [latex]f^{\prime}[/latex] is decreasing. We know that a differentiable function [latex]f^{\prime}[/latex] is decreasing if its derivative [latex]f^{\prime \prime}(x)<0[/latex]. Therefore, a twice-differentiable function [latex]f[/latex] is concave down when [latex]f^{\prime \prime}(x)<0[/latex]. Applying this logic is known as the concavity test.

Test for Concavity

Let [latex]f[/latex] be a function that is twice differentiable over an interval [latex]I[/latex].

  1. If [latex]f^{\prime \prime}(x)>0[/latex] for all [latex]x \in I[/latex], then [latex]f[/latex] is concave up over [latex]I[/latex].
  2. If [latex]f^{\prime \prime}(x)<0[/latex] for all [latex]x \in I[/latex], then [latex]f[/latex] is concave down over [latex]I[/latex].

We conclude that we can determine the concavity of a function [latex]f[/latex] by looking at the second derivative of [latex]f[/latex]. In addition, we observe that a function [latex]f[/latex] can switch concavity (Figure 6). However, a continuous function can switch concavity only at a point [latex]x[/latex] if [latex]f^{\prime \prime}(x)=0[/latex] or [latex]f^{\prime \prime}(x)[/latex] is undefined. Consequently, to determine the intervals where a function [latex]f[/latex] is concave up and concave down, we look for those values of [latex]x[/latex] where [latex]f^{\prime \prime}(x)=0[/latex] or [latex]f^{\prime \prime}(x)[/latex] is undefined. When we have determined these points, we divide the domain of [latex]f[/latex] into smaller intervals and determine the sign of [latex]f^{\prime \prime}[/latex] over each of these smaller intervals. If [latex]f^{\prime \prime}[/latex] changes sign as we pass through a point [latex]x[/latex], then [latex]f[/latex] changes concavity. It is important to remember that a function [latex]f[/latex] may not change concavity at a point [latex]x[/latex] even if [latex]f^{\prime \prime}(x)=0[/latex] or [latex]f^{\prime \prime}(x)[/latex] is undefined. If, however, [latex]f[/latex] does change concavity at a point [latex]a[/latex] and [latex]f[/latex] is continuous at [latex]a[/latex], we say the point [latex](a,f(a))[/latex] is an inflection point of [latex]f[/latex].

Definition

If [latex]f[/latex] is continuous at [latex]a[/latex] and [latex]f[/latex] changes concavity at [latex]a[/latex], the point [latex](a,f(a))[/latex] is an inflection point of [latex]f[/latex].

A sinusoidal function is shown that has been shifted into the first quadrant. The function starts decreasing, so f’ < 0 and f’’ > 0. The function reaches the local minimum and starts increasing, so f’ > 0 and f’’ > 0. It is noted that the slope is increasing for these two intervals. The function then reaches an inflection point (a, f(a)) and from here the slop is decreasing even though the function continues to increase, so f’ > 0 and f’’ < 0. The function reaches the maximum and then starts decreasing, so f’ < 0 and f’’ < 0.

Figure 6. Since [latex]f^{\prime \prime}(x)>0[/latex] for [latex]x<a[/latex], the function [latex]f[/latex] is concave up over the interval [latex](−\infty,a)[/latex]. Since [latex]f^{\prime \prime}(x)<0[/latex] for [latex]x>a[/latex], the function [latex]f[/latex] is concave down over the interval [latex](a,\infty)[/latex]. The point [latex](a,f(a))[/latex] is an inflection point of [latex]f[/latex].

Example: Testing for Concavity

For the function [latex]f(x)=x^3-6x^2+9x+30[/latex], determine all intervals where [latex]f[/latex] is concave up and all intervals where [latex]f[/latex] is concave down. List all inflection points for [latex]f[/latex]. Use a graphing utility to confirm your results.

Watch the following video to see the worked solution to Example: Testing for Concavity.

Try It

For [latex]f(x)=−x^3+\frac{3}{2}x^2+18x[/latex], find all intervals where [latex]f[/latex] is concave up and all intervals where [latex]f[/latex] is concave down.

We now summarize, in the box below, the information that the first and second derivatives of a function [latex]f[/latex] provide about the graph of [latex]f[/latex], and illustrate this information in Figure 8.

What Derivatives Tell Us about Graphs
Sign of [latex]f^{\prime}[/latex] Sign of [latex]f^{\prime \prime}[/latex] Is [latex]f[/latex] increasing or decreasing? Concavity
Positive Positive Increasing Concave up
Positive Negative Increasing Concave down
Negative Positive Decreasing Concave up
Negative Negative Decreasing Concave down
A function is graphed in the first quadrant. It is broken up into four sections, with the breaks coming at the local minimum, inflection point, and local maximum, respectively. The first section is decreasing and concave up; here, f’ < 0 and f’’ > 0. The second section is increasing and concave up; here, f’ > 0 and f’’ > 0. The third section is increasing and concave down; here, f’ > 0 and f’’ < 0. The fourth section is increasing and concave down; here, f’ < 0 and f’’ < 0.

Figure 8. Consider a twice-differentiable function [latex]f[/latex] over an open interval [latex]I[/latex]. If [latex]f^{\prime}(x)>0[/latex] for all [latex]x \in I[/latex], the function is increasing over [latex]I[/latex]. If [latex]f^{\prime}(x)<0[/latex] for all [latex]x \in I[/latex], the function is decreasing over [latex]I[/latex]. If [latex]f^{\prime \prime}(x)>0[/latex] for all [latex]x \in I[/latex], the function is concave up. If [latex]f^{\prime \prime}(x)<0[/latex] for all [latex]x \in I[/latex], the function is concave down on [latex]I[/latex].