Given `f'(1) = 1 and f(2x) = f(x) AA x gt 0`. If f'(x) is differentiable then prove that there exists a number `c in (2, 4)` such that f''(c) = - 1/8

If f(x) is continuous in [a, b] and differentiable in (a, b), then prove that there exists at least one c in (a, b) "such that" (f'(c))/(3c^(2)) = (f(b) - f(a))/(b^(3) - a^(3)) .

Let f(x) and g(x) be differentiable for 0 le x le 2 such that f(0) = 2, g(0) = 1, and f(2) = 8. Let there exist a real number c in [0, 2] such that f'(c) = 3g'(c) . Then find the value of g(2).

Let f (x) and g (x) be two differentiable functions for 0 le x le 1 such that f (0) = 2, g (0) = 0,f (1) = 6. If there exists a real number c in (0,1) such that f'(c ) = 2 g '(c ) , then g (1) is equal to

If f (x) = (x +2)/( 3x -1) , then f {f (x) } is

If f is differentiable at x=1, then lim_(x rarr 1) (x^(2) f(1)-f(x))/(x-1) is