suppose that 3 <_ f prime of x <_ 5, for all values x. show that 18<_ f(8)-f(2) <_ 30

Question

<_ signs mean less or equal to...

im supposed to apply mean value theorem or rolle's theorem... i don't understand neither so i cant do the question! please help!

Answer 1

If f'=3, then f=3x+C, so f(8)]=24+C, but C=2 if f'=3. So f(8)=26

f(2)=8 by the same argument.

then if f'x=3, f(8)-f(2)=26-8=>18

You can make the same logic if f'=5, solve for C, then show that f(8)-f(2)<=30