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)).

Expert Answers

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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.

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