Question

The polynomial function f(x) is an odd function. If x=c is a relative maximum of f(x), which of the following statements about c must be true?

A) c is a root of the polynomial.
B) c is an inflection point of the polynomial.
C) f(c)=0
D) f′(c)=0

Answer 1

If the polynomial function f(x) is an odd function and x=c is a relative maximum of f(x), then c must be a root of the polynomial and f(c)=0. Additionally, f'(c)=0 at x=c.

Step-by-step explanation:

If the polynomial function f(x) is an odd function and x=c is a relative maximum of f(x), then the following statements about c must be true:

A) c is a root of the polynomial.

Since f(x) is an odd function, it has symmetry about the origin. Therefore, if c is a relative maximum, then -c must also be a relative maximum. In order for c to be a root, it must be such that f(c) = 0.

C) f(c)=0

D) f'(c)=0

At a relative maximum or minimum, the derivative of the function is zero. Therefore, f'(c)=0 at x=c.