Question

Suppose that 4 ≤ f '(x) ≤ 5 for all values of x. what are the minimum and maximum possible values of f(9) − f(4)? ≤ f(9) − f(4) ≤

Answer 1

Assuming $f$ is differentiable everywhere, then by the mean value theorem, there is some $4<c<9$ such that
$f'(c)=\dfrac{f(9)-f(4)}{9-4}\implies 5f'(c)=f(9)-f(4)$

Since $4\le f'(x)\le5$,

$\implies f(9)-f(4)=5f'(c)\le5\cdot5=25$

$\implies f(9)-f(4)=5f'(c)\ge5\cdot4=20$

So $20\le f(9)-f(4)\le25$.