Suppose that 5 ≤ f '(x) ≤ 9 for all values of x. Show that 45 ≤ f(12) - f(3) ≤ 81. if the mean value theorem is used, justify that the conditions are satisfied

Question

Suppose that 5 ≤ f '(x) ≤ 9 for all values of x. Show that 45 ≤ f(12) - f(3) ≤ 81.

if the mean value theorem is used, justify that the conditions are satisfied

Answer 1

Since 5 ≤ f'(x) ≤ 9 for all x, f(x) is differentaible and continuous for all x (differentiability implies continuity)

So, f(x) is continuous on [3,12] and is differentiable on (3,12). Thus, by the Mean Value Theorem, there is at least one number c in (3,12) such that f'(c) = [f(12) - f(3)] / (12 - 3).

So, f(12) - f(3) = 9f'(c)

But, 5 ≤ f'(c) ≤ 9

So, 45 ≤ 9f'(c) ≤ 81

Therefore, 45 ≤ f(12) - f(3) ≤ 81