Let f be a differentiable function on `[0, 1]` s.t. `f(0) = f(1) = 0` and `f(1/4)=f(3/4)=1`. Show that `c_1, c_2, c_3` in (0, 1) s.t. `f'(c_1) + f'(c_2) + f'(c_3) = 0`.

Let f and g be differentiable on [0,1] such that f(0)=2,g(0),f(1)=6 and g(1)=2. Show that there exists c in(0,1) such that f'(c)=2g'(c)

If f and g are differentiable functions in [0, 1] satisfying f(0)""=""2""=g(1),g(0)""=""0 and f(1)""=""6 , then for some c in ]0,""1[ (1) 2f^'(c)""=g^'(c) (2) 2f^'(c)""=""3g^'(c) (3) f^'(c)""=g^'(c) (4) f'(c)""=""2g'(c)

If f and g are differentiable functions in [0, 1] satisfying f(0)""=""2""=g(1),g(0)""=""0 and f(1)""=""6 , then for some c in ]0,""1[ (1) 2f^'(c)""=g^'(c) (2) 2f^'(c)""=""3g^'(c) (3) f^'(c)""=g^'(c) (4) f'(c)""=""2g'(c)

If f and g are differentiable functions in [0, 1] satisfying f(0)""=""2""=g(1),g(0)""=""0 and f(1)""=""6 , then for some c in ]0,""1[ (1) 2f^'(c)""=g^'(c) (2) 2f^'(c)""=""3g^'(c) (3) f^'(c)""=g^'(c) (4) f'(c)""=""2g'(c)

If f and g are differentiable functions in [0, 1] satisfying f(0)""=""2""=g(1),g(0)""=""0 and f(1)""=""6 , then for some c in [0,""1] (1) 2f^'(c)""=g^'(c) (2) 2f^'(c)""=""3g^'(c) (3) f^'(c)""=g^'(c) (4) f'(c)""=""2g'(c)

If f and g are differentiable functions in [0, 1] satisfying f(0)""=""2""=g(1),g(0)""=""0 and f(1)""=""6 , then for some c in ]0,""1[ (1) 2f^'(c)""=g^'(c) (2) 2f^'(c)""=""3g^'(c) (3) f^'(c)""=g^'(c) (4) f'(c)""=""2g'(c)

Let f(x) and g(x) be differentiable functions for 0 le x le 1 such that f(0) = 2, g(0) = 0, f(1) = 6 . Let there exist a real number c in (0,1) such that f ’(c) = 2 g ’(c), then g(1) =

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

f(x) and g(x) are differentiable functions for 0 <= x <= 2 such that f(0) = 5, g(0) = 0, f(2)= 8,g(2) = 1. Show that there exists a number c satisfying 0 < c < 2 and f'(c)=3 g'(c).

If f and g are differentiable functions for 0 le x le 1 such that f(0) = 2, g(0), f (1) = 6, g (1) = 2 , then show that there exists c satisying 0 < c < 1 and f' (c ) = 2g' ( c)