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))`.

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 y = f(x) is continuous on [0, 6] differentiable on (0, 6), f(0) = -2 and f(6) = 16, then at some point between x = 0 and x = 6, f'(x) must be equal to a) -18 b) -3 c)3 d)14

If f is defined and continuous on [3, 5]and f is differentiable at x=4 and f' (4) = 6 , then the value of underset(x to 0)(lim) (f(4 + x) - f(4- x) )/( 4x) is equal to a)0 b)2 c)3 d)4

Let f be continuous on [1,5] and differentiable in (1,5) .if f(1) =-3 and f' (x) ge 9 for all x in (1,5) then : a) f (5) , ge 33 b) f (5) ge 36 c) f (5) le 36 d) f (5) ge 9

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x), g'(x) be a, b, respectively, (a lt b) . Show that there exists at least one root of equation f'(x) g'(x) + f(x) g''(x) = "0 on" (a, b) .