4.5 Derivatives and the Shape of a Graph

Earlier in this chapter we stated that if a function $f$ has a local extremum at a point $c,$ then $c$ must be a critical point of $f .$ However, a function is not guaranteed to have a local extremum at a critical point. For example, $f (x) = x^{3}$ has a critical point at $x = 0$ since $f^{'} (x) = 3 x^{2}$ is zero at $x = 0,$ but $f$ does not have a local extremum at $x = 0.$ Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.

The First Derivative Test

Corollary $3$ of the Mean Value Theorem showed that if the derivative of a function is positive over an interval $I$ then the function is increasing over $I .$ On the other hand, if the derivative of the function is negative over an interval $I,$ then the function is decreasing over $I$ 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 4.30Both functions are increasing over the interval $(a, b) .$ At each point $x,$ the derivative $f^{'} (x) > 0.$ Both functions are decreasing over the interval $(a, b) .$ At each point the derivative

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

Note that $f$ need not have local extrema at a critical point. The critical points are candidates for local extrema only. In Figure 4.31, we show that if a continuous function $f$ 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 $f$ has a local extremum at a critical point, then the sign of $f^{'}$ switches as $x$ 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 4.31The function $f$ has four critical points: $a, b, c, and d .$ The function $f$ has local maxima at $a$ and $d,$ and a local minimum at $b .$ The function does not have a local extremum at The sign of changes at all local extrema.

Using Figure 4.31, we summarize the main results regarding local extrema.

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

This result is known as the first derivative test.

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

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

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 4.34(a) shows a function $f$ with a graph that curves upward. As $x$ increases, the slope of the tangent line increases. Thus, since the derivative increases as $x$ increases, $f^{'}$ is an increasing function. We say this function $f$ is concave up. Figure 4.34(b) shows a function $f$ that curves downward. As $x$ increases, the slope of the tangent line decreases. Since the derivative decreases as $x$ increases, $f^{'}$ is a decreasing function. We say this function is concave down.

In general, without having the graph of a function $f,$ how can we determine its concavity? By definition, a function $f$ is concave up if $f^{'}$ is increasing. From Corollary $3,$ we know that if $f^{'}$ is a differentiable function, then $f^{'}$ is increasing if its derivative Therefore, a function that is twice differentiable is concave up when Similarly, a function is concave down if is decreasing. We know that a differentiable function is decreasing if its derivative Therefore, a twice-differentiable function is concave down when Applying this logic is known as the .

We conclude that we can determine the concavity of a function $f$ by looking at the second derivative of $f .$ In addition, we observe that a function $f$ can switch concavity (Figure 4.35). However, a continuous function can switch concavity only at a point $x$ if $f^{''} (x) = 0$ or $f^{''} (x)$ is undefined. Consequently, to determine the intervals where a function is concave up and concave down, we look for those values of where or is undefined. When we have determined these points, we divide the domain of into smaller intervals and determine the sign of over each of these smaller intervals. If changes sign as we pass through a point then changes concavity. It is important to remember that a function may not change concavity at a point even if or is undefined. If, however, does change concavity at a point and is continuous at we say the point is an inflection point of

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 4.35Since $f^{''} (x) > 0$ for $x < a,$ the function $f$ is concave up over the interval $(- \infty, a) .$ Since for the function is concave down over the interval The point is an inflection point of

We now summarize, in Table 4.1, the information that the first and second derivatives of a function $f$ provide about the graph of $f,$ and illustrate this information in Figure 4.37.

Table 4.1What Derivatives Tell Us about Graphs

Sign of $f^{'}$	Sign of $f^{''}$	Is $f$ increasing or decreasing?	Concavity
Positive	Positive	Increasing	Concave up
Positive	Negative	Increasing

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 4.37Consider a twice-differentiable function $f$ over an open interval $I .$ If $f^{'} (x) > 0$ for all $x \in I,$ the function is increasing over If for all the function is decreasing over If for all the function is concave up. If for all the function is concave down on

The Second Derivative Test

The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.

We know that if a continuous function has local extrema, it must occur at a critical point. However, a function need not have local extrema at a critical point. Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Let $f$ be a twice-differentiable function such that $f^{'} (a) = 0$ and $f^{''}$ is continuous over an open interval $I$ containing $a .$ Suppose $f^{''} (a) < 0.$ Since is continuous over for all (Figure 4.38). Then, by Corollary is a decreasing function over Since we conclude that for all if and if Therefore, by the first derivative test, has a local maximum at On the other hand, suppose there exists a point such that but Since is continuous over an open interval containing then for all (Figure 4.38). Then, by Corollary is an increasing function over Since we conclude that for all if and if Therefore, by the first derivative test, has a local minimum at

A function f(x) is graphed in the first quadrant with a and b marked on the x-axis. The function is vaguely sinusoidal, increasing first to x = a, then decreasing to x = b, and increasing again. At (a, f(a)), the tangent is marked, and it is noted that f’(a) = 0 and f’’(a) < 0. At (b, f(b)), the tangent is marked, and it is noted f’(b) = 0 and f’’(b) > 0. — Figure 4.38Consider a twice-differentiable function $f$ such that $f^{''}$ is continuous. Since $f^{'} (a) = 0$ and $f^{''} (a) < 0,$ there is an interval containing such that for all in is increasing if and is decreasing if As a result, has a local maximum at Since and there is an interval containing such that for all in is decreasing if and is increasing if As a result, has a local minimum at

Note that for case iii. when $f^{''} (c) = 0,$ then $f$ may have a local maximum, local minimum, or neither at $c .$ For example, the functions $f (x) = x^{3},$ $f (x) = x^{4},$ and all have critical points at In each case, the second derivative is zero at However, the function has a local minimum at whereas the function has a local maximum at and the function does not have a local extremum at

Let’s now look at how to use the second derivative test to determine whether $f$ has a local maximum or local minimum at a critical point $c$ where $f^{'} (c) = 0.$

We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as $x \to \pm \infty.$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.

Key Concepts

If $c$ is a critical point of $f$ and $f^{'} (x) > 0$ for $x < c$ and $f^{'} (x) < 0$ for then has a local maximum at