Proof using mean-value theorem

[javi438]

use the mean-value theorem to show that if f is continuous at x and at x+h and is differentiable between these 2 numbers, then f(x+h) - f(x) = f'(x+ah)h for some number a between 0 and 1.

mvt: if f is diff'ble on (a,b) and continuous on [a,b] then there is at least one number c in (a,b) for which f'(c)=[f(b)-f(a)]/(b-a)

any help will be appreciateddd..i don't know where to start :(

 

[Skins]

Try thinking of the expression

f(x+h) = f(x) = hf'(x+ah) where 0 < a < 1

as another form of expressing the mean value theorem . You know that f is continuous on [x, x+h] and differentiable on (x, x+h). So now apply the MVT. I don't want to give much more info yet because I'd be giving up the whole proof. Give it a try and see how far you can get with it.