Question

Suppose that f(x) is continuous and differentiable on the interval [2,5]. Suppose also that f(2) = 9 and f’(x) ≤9. What is the largest possible value for f(5)?

Answer 1

We'll try to use the definition of the derivative:
f'(x) = (f(b) - f(a))/(b-a))
thus
(f(5) - f(2))/(5-2) = (f(5) - 9) /3 ≤ 9.
(f(5) - 9) ≤ 27
f(5) ≤ 36.
Answer: 36.