State whether the following statement is true or false. If it is false, correct the statement and...

Question:

State whether the following statement is true or false. If it is false, correct the statement and explain why it is not true, or give a counter example.

If {eq}f''(c) = 0 {/eq} at a point {eq}x=c {/eq}, then {eq}c {/eq} is the point of inflection.

Inflection Points:

Let {eq}f(x) {/eq} be a function. We say that the point {eq}x=c {/eq} is an inflection point of {eq}f(x) {/eq} if {eq}f(x) {/eq} changes concavity at {eq}x=c {/eq}. That is, {eq}c {/eq} is an inflection point if there is some {eq}r {/eq} such that either:

1) {eq}f(x) {/eq} is concave down on the interval {eq}(c-r,c] {/eq} and concave up on the interval {eq}[c,c+r) {/eq}, or

2) {eq}f(x) {/eq} is concave up on the interval {eq}(c-r,c] {/eq} and concave down on the interval {eq}[c,c+r) {/eq}.

Answer and Explanation:

This statement is false. Let {eq}f(x)=x^4 {/eq}. Then differentiating gives {eq}f'(x)=4x^3 {/eq} and so {eq}f''(x)=12x^2 {/eq}. Thus {eq}f''(0)=0 {/eq}.

However, {eq}f''(x)>0 {/eq} whenever {eq}x \ne 0 {/eq}, and so {eq}f(x) {/eq} is concave up both on the interval {eq}(-\infty, 0] {/eq} and on the interval {eq}[0,\infty) {/eq}. So {eq}0 {/eq} is not an inflection point of {eq}f(x) {/eq}.


Learn more about this topic:

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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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