Question

Suppose a local maximum occurs at x=c for a function . which of the following must be true? select all that apply.

a. f'(c) = 0
b. f""(c) > 0
c. f(c) is defined
d. F(x) is continous at x = c

Answer 1

A local maximum occurs at x=c for a function. Statements a and c must be true, while statements b and d may or may not be true.

Step-by-step explanation:

To determine which statements are true when a local maximum occurs at x=c for a function, we need to consider the properties of the function at that point.

a. f'(c) = 0: This statement must be true. At a local maximum, the derivative of the function is zero.

b. f"(c) > 0: This statement may or may not be true. The second derivative of the function tells us about concavity, not the presence of a local maximum.

c. f(c) is defined: This statement must be true. The function must be defined at the point of local maximum.

d. F(x) is continuous at x=c: This statement may or may not be true. The function can have a local maximum even if it is not continuous at that point.