Suppose that f(0)=1 and f '(x)≤5 for all values of x. Use the Mean Value Theorem to determine how large f(4) can possibly be.

Question

Suppose that f(0)=1 and f '(x)≤5 for all values of x. Use the Mean Value Theorem to determine how large f(4) can possibly be.

Answer f(4)≤____

Answer 1

Let us apply the mean value theorem in the interval [0,4]. There exists a constant c in (0,4) such that f'(c)=(f(4)-f(0))/(4-0)=(f(4)-1)/4. Since for all values of x we have that f'(x)<=5 we immediately obtain that (f(4)-1)/4<=5 which implies that f(4)-1<=20 or equivalently, f(4)<=21.