1. 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 ƒ, if ƒ"(0) = 0, there is an inflection point at x = 0.

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

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

(d) For ________ functions ƒ, if ƒ' is continuous for all real numbers and ƒ has no critical points, then f is everywhere increasing or everywhere decreasing.

In mathematics, we consider a statement to be false if we can find any examples where the statement is not true. We refer to these examples as counterexamples. Note that a counter example is an example for which the "if" part of the statement is true, but the "then"part of the statement is false.