Question

Suppose that f(0)=3 and f′(x)≤4 for all values of x. Use the Mean Value Theorem to determine how large f(4) can possibly be.

Answer 1

What happened when you tried to apply the M.V.T. ?

Answer 2

by MVT we have

[f(4) - f(0)]/ ( 4 - 0) = f' (c)

Answer 3

Let f ' (x) be the largest it can be. So let f ' (c) = 4 for some value of c

Also, let f(4) = x

By the mean value theorem, we know

\[\Large \frac{f(4)-f(0)}{4-0} = f \ '(c)\]

\[\Large \frac{x-3}{4-0} = 4\]

solve for x

Answer 4

f(4) - 3 = f ' (c) * 4
f(4) = f ' (c) *4 + 3

now you are given that f ' (x) <= 4 for all x.
this means

f(4) <= 4 * 4 + 3 , by substitution