Question

3. 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 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”. An example must include either a graph or a specific function. (a) For functions f, if f has an antiderivative, then this antiderivative is unique. (b) For functions f, if f(x) is a linear function with positive slope, then any antiderivative of f is also a linear function. (c) For antiderivatives F of a given function f, if f(x) is positive over an interval, then F(x) is also positive over that interval. (d) For functions f, if f(c) is increasing over an interval, then every antiderivative F(x) is concave up over that interval. 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 counterexample is an example for which the “if” part of the statement is true, but the “then” part of the statement is false.

Answer 1

But But take take ma max f) ma max fox) a6 then then ) " aaliderivative & F(X) = m f(a) which is an antidesiva negative over that laro of is (d) all O ms of lo va Because we know when the function y = g(x) is conceare up, the graph its derivative y = f(x) is increasing