# Suppose you are given a formula for a function f.Suppose you are given a formula for a function f. (c) How do you locate inflection points? a) At any value of x where the concavity changes, we have...

Suppose you are given a formula for a function *f*.

Suppose you are given a formula for a function *f*.

(c) How do you locate inflection points?

a) At any value of *x* where the concavity changes, we have an inflection point at (*x*, *f*(*x*)).

b) At any value of *x* where *f'*(*x*) = 0, we have an inflection point at (*x*, *f*(*x*)).

c) At any value of *x* where the concavity does not change, we have an inflection point at (*x*, *f*(*x*)).

d) At any value of *x* where the function changes from increasing to decreasing, we have an inflection point at (*x*, *f*(*x*)).

e) At any value of *x* where the function changes from decreasing to increasing, we have an inflection point at (*x*, *f*(*x*)).

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### 1 Answer

**A point of inflection is located where the concavity of the function changes, so the answer is (a).**

(b) The first derivative equals zero could be an inflection point, but it doesn't have to be. Consider the vertex of a parabola.

(c) Just because the second derivative is zero doesn't mean you have an inflection point. Consider `y=x^4` at x=0. The concavity does not change but the second derivative is zero.

(d) If the function changes from increasing to decreasing there is a local maximum (assuming the function is continuous), but that isn't necessarily an inflection point. Consider a parabola opening down.

(e) Same as (d); consider a parabola opening up. The point is a local minimum, not necessarily an inflection point.