Suppose that f and g are continuous on [a,b] and differentiable on (a,b) and suppose that f(a)=g(a) and f'(x)

Question

Suppose that f and g are continuous on [a,b] and differentiable on (a,b) and suppose that f(a)=g(a) and f'(x) < g'(x) for a<x<b. 

Prove that f(b)<g(b)

paxpolaris · Answer

are you familiar with  mean value theorem ?

anonymous · Answer

yes

anonymous · Answer

if f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b)  there there is at least one number c in (a,b)  such that 

f'(c) = f(b)-f(a)/b-a

paxpolaris · Answer

nvm .. you can't actually use it.

anonymous · Answer

actually i think you can?

anonymous · Answer

wait no... i dont know what im talking about

paxpolaris · Answer

how about: $$f'(x)<g'(x)$$

can we take the definite integral from a to b....

paxpolaris · Answer

can we do this:$$\int\limits_a^b f'(x) < \int\limits_a^b g'(x)$$

anonymous · Answer

thanks for your help

anonymous · Answer

I don't think integrals are necessary here. You're explicitly told that $f'(x)<g'(x)$, so $f$ is decreasing faster (if both derivatives are negative) or increasing slower (if both are positive) than $g$, and so $f(b)<g(b)$ provided that $a<b$.

anonymous · Answer

One more case to consider, actually, however trivial it may seem. If $f'<0$ and $g'>0$, then $g$ is increasing faster than $f$, so again $f(b)<g(b)$.