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Why is the second derivative of an inflection point zero?

Question:

Why is the second derivative of an inflection point zero?

Inflection Points and Concavity

The graph of a function {eq}f(x) {/eq} is concave up (curved like a smile) when {eq}f''(x)>0 {/eq} and is concave down (curved like a frown) when {eq}f''(x)<0. {/eq} An inflection point is a point on the graph of {eq}f(x) {/eq} where the graph changes from concave up to concave down or from concave down to concave up

Answer and Explanation:

An inflection point is a point on the graph of {eq}f(x) {/eq} where the graph changes concavity. The graph is concave up when {eq}f''(x) >0 {/eq} and concave down when {eq}f''(x)<0. {/eq} A point where the concavity changes would then need to be a point where the second derivative is neither positive nor negative. This means that either {eq}f''(x) = 0 {/eq} or {eq}f''(x) {/eq} is undefined at any inflection point.


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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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