Question

. If f is differentiable and f(-4) = f(4), then there is a number c such that |c| < 4 and f'(c) = ____.

a) 0

b) 1

c) 4

d) -4

Answer 1

If a function is differentiable and f(-4) = f(4), then there exists a number c such that |c| < 4 and f'(c) = 0.

Step-by-step explanation:

If a function f is differentiable and f(-4) = f(4), then there is a number c such that |c| < 4 and f'(c) = 0.

This is based on the Mean Value Theorem which states that if a function is continuous on a closed interval (in this case, [-4, 4]) and differentiable on the open interval (in this case, (-4, 4)), then there exists a number c in the open interval such that f'(c) is equal to the average rate of change of the function over the closed interval. Since f(x) = f(-4) = f(4), the average rate of change of the function is 0, and therefore, there exists a number c such that f'(c) = 0.