Derivatives and the Shape of a Graph: Learn It 2

Lumen Learning

Derivatives and the Shape of a Graph: Learn It 2

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.

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].

concave up and concave down

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].

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].

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].
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.

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].

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].

inflection point

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].

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.

To determine concavity, we need to find the second derivative [latex]f^{\prime \prime}(x)[/latex].

The first derivative is [latex]f^{\prime}(x)=3x^2-12x+9[/latex], so the second derivative is [latex]f^{\prime \prime}(x)=6x-12[/latex].

If the function changes concavity, it occurs either when [latex]f^{\prime \prime}(x)=0[/latex] or [latex]f^{\prime \prime}(x)[/latex] is undefined. Since [latex]f^{\prime \prime}[/latex] is defined for all real numbers [latex]x[/latex], we need only find where [latex]f^{\prime \prime}(x)=0[/latex].

Solving the equation [latex]6x-12=0[/latex], we see that [latex]x=2[/latex] is the only place where [latex]f[/latex] could change concavity.

We now test points over the intervals [latex](−\infty ,2)[/latex] and [latex](2,\infty)[/latex] to determine the concavity of [latex]f[/latex]. The points [latex]x=0[/latex] and [latex]x=3[/latex] are test points for these intervals.

Interval	Test Point	Sign of [latex]f^{\prime \prime}(x)=6x-12[/latex] at Test Point	Conclusion
[latex](−\infty ,2)[/latex]	[latex]x=0[/latex]	[latex]-[/latex]	[latex]f[/latex] is concave down
[latex](2,\infty )[/latex]	[latex]x=3[/latex]	[latex]+[/latex]	[latex]f[/latex] is concave up.

We conclude that [latex]f[/latex] is concave down over the interval [latex](−\infty ,2)[/latex] and concave up over the interval [latex](2,\infty)[/latex]. Since [latex]f[/latex] changes concavity at [latex]x=2[/latex], the point [latex](2,f(2))=(2,32)[/latex] is an inflection point. Figure 7 confirms the analytical results.

The function f(x) = x3 – 6x2 + 9x + 30 is graphed. The inflection point (2, 32) is marked, and it is roughly equidistant from the two local extrema. — Figure 7. The given function has a point of inflection at [latex](2,32)[/latex] where the graph changes concavity.

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the transcript for this segmented clip of “4.5 Derivatives and the Shape of a Graph” here (opens in new window).

The table and figure below summarize how the first and second derivatives of a function [latex]f(x)[/latex] inform the characteristics of its graph.

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].