Suppose that f and g are continuous on [a, b]anddifferentiable on (a, b). Suppose also that f(a) = g(a) andf'(x)g'(x) for axb.Prove that f(b) g(b).
Accepted Solution
A:
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 thatf(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[tex]\dfrac{p^\prime(x)}{q^\prime(x)}=\dfrac{p(b)-p(a)}{q(b)-q(a)}.[/tex]Since f(x) and g(x) satisfies the conditions of Cauchy Mean Value Theorem, so we get[tex]\dfrac{f^\prime(x)}{g^\prime(x)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}\\\\\\\Rightarrow \dfrac{f^\prime(x)}{f^\prime(x)}=\dfrac{f(b)-f(a)}{g(b)-f(a)}~~~~~~~~~[\textup{since }f(a)=g(a)~\textup{and }f^\prime(x)=g^\prime(x)]\\\\\\\Rightarrow 1=\dfrac{f(b)-f(a)}{g(b)-f(a)}\\\\\Rightarrow f(b)-f(a)=g(b)-f(a)\\\\\Rightarrow f(a)=g(a).[/tex]Thus, f(b) = g(b).Hence proved.