MATH 12002 - CALCULUS I §3.3: Concavity & Inflection Points

Professor Donald L. White

Department of Mathematical Sciences Kent State University

D.L. White (Kent State University) 1 / 5 Concavity

Definition A function f is concave up on an interval if the graph of f lies above its tangent lines on the interval:

¡ @ ¡ @ CONCAVE UP & % A function f is concave down on an interval if the graph of f lies under its tangent lines on the interval:

¡ @ ¡ @ ¡' $ CONCAVE DOWN An inflection point of f (or point of inflection, if you’re not into the whole brevity thing) is a point on the graph of f where f changes concavity from up to down or from down to up.

D.L. White (Kent State University) 2 / 5 Concavity

We can determine the concavity of f on an interval by finding the sign of the second derivative f 00 of f . Notice that if f is concave up, the slopes of the tangent lines increase from left to right:

¡ @ 2 −1 ¡ @ 0 &CONCAVE UP % Thus f 0 is increasing on the interval, and its derivative, f 00, is positive. But if f is concave down, the slopes of the tangent lines decrease from left to right: 0 @ −1 2 ¡ ¡ @ ¡' $

CONCAVE DOWN Thus f 0 is decreasing on the interval, and its derivative, f 00, is negative.

D.L. White (Kent State University) 3 / 5 Concavity

We therefore have: Concavity Test Let y = f (x) be a function. If f 00(x) > 0 on an interval I , then f is concave up on I . If f 00(x) < 0 on an interval I , then f is concave down on I .

Now f 00 can only change sign at a number c where f 00(c) = 0 or f 00(c) is undefined. Since f 00 is the derivative of f 0, such a number is a critical number for f 0. Second Derivative Test for Inflection Point Let y = f (x) be a function defined at x = c, where c is a number such that f 00(c) = 0 or f 00(c) is undefined. If f 00(x) changes sign at x = c, then f has an inflection point at c.

D.L. White (Kent State University) 4 / 5 Concavity

As we noted previously, a function g that is continuous on its domain can change sign only at a zero of g or at a point where g is undefined.

To graph a function y = f (x) it will therefore be useful to find all numbers where at least one of f 0 or f 00 is zero or undefined. Between any two such numbers, neither f 0 nor f 00 can change sign, and so f has one of the following shapes on such an interval:

' $ f 0 + f 0 + f 0 − f 0 − f 00 + %f 00 − &f 00 + f 00 −

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