Answer to Question #198721 in Calculus for desmond

Question

1.Suppose that f and g are continuous on [a, b] and differentiable on (a, b). Suppose also that f(a) = g(a) and f 0 (x) < g0 (x) for a < x < b. Prove that f(b) < g(b).

Answer 1

In the case of continuous and differentiable functions f(x), g(x) we have a definition of differential:

$f(x)-f(\xi) = f'(\xi)(x-\xi)+o(x)$

$g(x)-g(\xi) = g'(\xi)(x-\xi)+o(x)$

and finally

$f(x)-g(x) = (f'(\xi)-g'(\xi))(x-\xi)+o(x)$

We know that:

$\xi\in[a,b],x-\xi>0, f'(\xi)-g'(\xi)>0$

So this statement are equivalent to the next one:

$f(b)-g(b)>0$