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)=(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`.