Question

Let f ′be the function with a derivative given by f ′(x)=x2 − a2 =(x−a)(x+a), where a is a positive constant. Which of the following statements is true?

A) f ′(a)=0
B) f ′(a)=2a 2
C) f ′(x)=0
D) f ′(x)=2x 2

Answer 1

Upon substitution of x with a in the function f'(x) = x^2 - a^2, we get f'(a) = a^2 - a^2 = 0. Therefore, the correct statement about the derivative of the function when a is a positive constant is that f'(a) = 0. Option A is the correct answer.

Step-by-step explanation:

The question provides us with the derivative of a function f'(x) = x^2 - a^2, which can also be expressed as the product of two binomials (x - a)(x + a). Our goal is to evaluate the correctness of given statements about this derivative when a is a positive constant.

Statement A suggests f'(a) = 0. To verify this, we substitute x with a in the derivative: f'(a) = a^2 - a^2 = 0^2 = 0. This shows that Statement A is true. Statements B, C, and D can be considered incorrect without further calculation since A is true and the options are mutually exclusive in this context. Therefore, the correct option is A.