Question

Fill in the blank with “all”, “no”, or “some” to make the following statements true. Note that “some” means one or more instances, but not all. • If your answer is “all,” then give a brief explanation as to why. • If your answer is “no,” then give an example and a brief explanation as to why. • If your answer is “some,” then give two specific examples that illustrate why your answer it not “all” or “no.” Be sure to explain your two examples. An example must include either a graph or a specific function.

(a) For functions f, if f′′(0) = 0, there is an inflection point at x = 0. (

b) For functions f, if f′(p) = 0, then f has a local minimum or maximum at x = p.

(c) For functions f, a local minimum of a function f occurs at a critical point of f.

(d) For functions f, if f′ is continuous and f has no critical points, then f is everywhere increasing or everywhere decreasing.

Answer 1

(a) For SOME functions , if , there is an inflection point at .

First consider the function . We know that is an inflection point for . Therefore, the answer cannot be NO.

If we consider then and . But is not an inflection point since is concave upward in all its domain.

(b) For SOME functions , if , then f has a local minimum or maximum at .

Consider the function . Then is zero only if . And is a minimum for . Therefore, for any such that we have that is a minimum. We conclude that the answer cannot be NO.

Let . Then and for we have . But is nor a minimum nor a maximum for the function . Therefore, the answer cannot be ALL.

(c) For ALL functions , a local minimum of a function occurs at a critical point of .

Remember that a critical point of a function is a point where is not differentiable or . If we do not have the first case, that is, if is differentiable, the tangent line at a minimum is horizontal. Therefore, the derivate is zero.

(d) For ALL functions , if is continuous and has no critical points, then is everywhere increasing or everywhere decreasing.

Suppose that contradiction that the affirmation is false. So, there is a function that is locally increasing and locally decreasing in different regions of its domain. Then, the function must have either a maximum or a minimum between the two regions. This minimum/maximum corresponds to a critical poit.