JEE Advance - Mathematics (1982 - No. 38)
If $$f(x)$$ and $$g(x)$$ are differentiable function for $$0 \le x \le 1$$ such that $$f(0)=2$$, $$g(0)=0$$, $$f(1)=6$$; $$g(1)=2$$, then show that there exist $$c$$ satisfying $$0 < c < 1$$ and $$f'(c)=2g'(c)$$.
Apply the Mean Value Theorem to f(x) and g(x) separately.
Consider the function h(x) = f(x) - 2g(x) and apply Rolle's Theorem.
Consider the function h(x) = 2f(x) - g(x) and apply Rolle's Theorem.
Use Cauchy's Mean Value Theorem with f(x) and g(x).
The Mean Value Theorem is not applicable in this case.