Proof f(x)>g(x) in an interval

[SammyS]

Homework Statement

let fx, gx be continuous in [a,b] and differentiable in (a,b). at the end of the interval f(a) >= g(a).

and f'(x) >g'(x) for a<x<b.

proof f(x) > g(x) for a<x<=b

I that supposed to be

let f(x), g(x) be continuous in [a,b] ...​

?

Attempt:
There is a statement says that if the f'x = g'x for x in [a,b] , then there exists k such that f'x - g'x = k for any x in [a,b]

but f'(x) = g'(x) + t(x), where t(x) isn't have to be a line. and i cannot use the statement

Did you mean that literally?

There is a statement says that if the f'x = g'x for x in [a,b] , then there exists k such that f'x - g'x = k ...​

or perhaps:

f'(x) = g'(x) for x in [a,b] , then there exists k such that f'(x) - g'(x) = k ...​

.

You need to put a bit more effort in here! You're doing real analysis, which requires precision in terms of the statement of a problem. I'll help you on this point, but you need to start thinking more (pure) mathematically:

We have that $k$ is continuous on $[a, b]$, differentiable on $(a, b)$, $k(a) \ge 0$ and $\forall x \in (a, b), \ k'(x) > 0$.

We must show that $\forall x \in (a, b], \ k(x) > 0$

Thank you @PeroK !