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4.5 Derivatives and the shape of a graph

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  • 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.
  • Explain the relationship between a function and its first and second derivatives.
  • State the second derivative test for local extrema.

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.
Both 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 x , the derivative f ′ ( x ) < 0 .

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 c and differentiable over I , except possibly at c , the only way f can switch from increasing to decreasing (or vice versa) at point c is if f ′ changes sign as x increases through c . If f is differentiable at c , the only way that f ′ . can change sign as x increases through c is if f ′ ( c ) = 0 . Therefore, for a function f that is continuous over an interval I containing c and differentiable over I , except possibly at c , the only way f can switch from increasing to decreasing (or vice versa) is if f ′ ( c ) = 0 or f ′ ( c ) is undefined. Consequently, to locate local extrema for a function f , we look for points c in the domain of f such that f ′ ( c ) = 0 or f ′ ( c ) is undefined. Recall that such points are called critical points of f .

Note that f need not have a local extrema at a critical point. The critical points are candidates for local extrema only. In [link] , 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.
The 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 f does not have a local extremum at c . The sign of f ′ changes at all local extrema.
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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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