MATH 12002 - CALCULUS I 3.3: Concavity & Inflection Points - - PowerPoint PPT Presentation

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

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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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.

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Concavity

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

✫ ✪

CONCAVE UP

❅ ❅ ❅ ❅

−1

✁ ✁ ✁ ✁

2

Thus f ′ is increasing on the interval, and its derivative, f ′′, is positive. But if f is concave down, the slopes of the tangent lines decrease from left to right:

✬ ✩

CONCAVE DOWN

✁ ✁ ✁ ✁

2

❅ ❅ ❅ ❅ −1

Thus f ′ is decreasing on the interval, and its derivative, f ′′, is negative.

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Concavity

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.

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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 ′ 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 ′′ −

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