Suppose that f and g are continuous on [a, b]

anddifferentiable on (a, b). suppose also that f(a) = g(a) and

f'(x)

g'(x) for

axb.

prove that f(b)

g(b).

 The proof is given below.

Step-by-step explanation:  Given that f and g are continuous on [a, b] and differentiable on (a, b). Also, f(a) = g(a) and f'(x) = g'(x) for a ≠ b.

We are to prove that

f(b) = g(b).

Cauchy Mean Vale Theorem:   If p(x) and q(x) are any two functions that are continuous on [a, b] and differentiable on (a, b), then for some x in (a, b), we have

Since f(x) and g(x) satisfies the conditions of Cauchy Mean Value Theorem, so we get

Thus, f(b) = g(b).

Hence proved.