Answer to Question #198215 in Calculus for desmond

Question

3. 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

Given

f and g are continues in [a , b]

f and g are differentiable in (a,b)

$f(a)=g(a)$

and $f(x)<g(x)$

to prove $f(b)<g(b)$

as given , that the both function are continues in the closed interval a , b

and also differentiable in open interval a, b

it islo given that $f(x)<g(x)$ for every x given in the domain

but $f(a)=g(a)$ this means the starting point of both curve is same than after function g is above the function f so the value of functions at a particular point x is different and the value of function g is greater than the function f .

so we can say that at point b $f(b)<g(b)$

hence proved.