Why isn't it enough to conclude that f has an inflection point at (c,f(c)) when f''(c) = 0 ?

Question:

Why isn't it enough to conclude that {eq}f {/eq} has an inflection point at {eq}(c,f(c)) {/eq} when {eq}f''(c) = 0 {/eq}?

Concavity and Inflection Points

A common misconception is that a function has an inflection point whenever the second derivative is zero. However, the true statement is that the function has an inflection point whenever the second derivative crosses the x-axis, as this means that the concavity of the function changes.

Answer and Explanation:

Setting the second derivative equal to zero is a good first step to finding the inflection points for a function, but it is not the only step we need to take. This is because inflection points occur when concavity changes, so the second derivative must cross the x-axis. Thus, we need to verify that the second derivative does this.

Recall a function like {eq}f(x) = x^2 {/eq}. If this is the second derivative, we'd be inclined to say that the function has an inflection point at zero. However, notice that this function only touches the x-axis at this point instead of crossing it. This means that the original function would be entirely concave up and thus have no inflection points.

Thus, we need to do the same thing that we do for critical points and verify that the second derivative changes sign on either side of the possible inflection point. We can do this by testing points on either side or by checking that the third derivative is also not equal to zero at this point.


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Understanding Concavity and Inflection Points with Differentiation

from Math 104: Calculus

Chapter 10 / Lesson 6
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