Calculus I: Concavity and Inflection Points, Study notes of Calculus

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MATH 12002 - CALCULUS I

§3.3: Concavity & Inflection Points

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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.

We therefore have:

Concavity Test

Let y = f (x) be a function.

If f ′′(x) > 0 on an interval I , then f is concave up on I. If f ′′(x) < 0 on an interval I , then f is concave down on I.

Now f ′′^ can only change sign at a number c where f ′′(c) = 0 or f ′′(c) is undefined. Since f ′′^ is the derivative of f ′, such a number is a critical number for f ′.

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 ′′(c) = 0 or f ′′(c) is undefined. If f ′′(x) changes sign at x = c, then f has an inflection point at c.

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 ′^ or f ′′^ is zero or undefined.

Between any two such numbers, neither f ′^ nor f ′′^ can change sign, and so f has one of the following shapes on such an interval:

% f ′^ + f ′′^ +

'

f ′^ + f ′′^ −

& f ′^ − f ′′^ +

$

f ′^ − f ′′^ −