Question

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).

Answer 1

Answer: 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

`(p^\prime(x))/(q^\prime(x))=(p(b)-p(a))/(q(b)-q(a)).`

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

`(f^\prime(x))/(g^\prime(x))=(f(b)-f(a))/(g(b)-g(a))\\\\\\\Rightarrow (f^\prime(x))/(f^\prime(x))=(f(b)-f(a))/(g(b)-f(a))~~~~~~~~~[\textup{since }f(a)=g(a)~\textup{and }f^\prime(x)=g^\prime(x)]\\\\\\\Rightarrow 1=(f(b)-f(a))/(g(b)-f(a))\\\\\Rightarrow f(b)-f(a)=g(b)-f(a)\\\\\Rightarrow f(a)=g(a).`

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

Hence proved.