Discussion Forum



1)

 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 ε ]0,1[


A) $2f'(c)=g'(c)$

B) $2f'(c)=3g'(c)$

C) $f'(c)=g'(c)$

D) $f'(c)=2g'(c)$

Answer:

Option D

Explanation:

Given  , f(0)= 2=g(1),g(0)=0  and f(1)=6

  f and g are differentiable  in (0,1)

Let h(x)= f(x)-2g(x)  .......(i)

   h(0)=f(0)-2g(0)

 h(0)=2-0

    h(0)=2

 and h(1)=f(1)-2g(1)=6-2(2)

  h(1)=2, h(0)=h(1)=2

Hence, using Rolle's theorem.

    h'(c)=0,  such that c ε  (0,1)

 Differentiating Eq.(i) at, we get

 $\Rightarrow$    f'(c)-2 g'(c)=0

 $\Rightarrow$  f '(c) =2 g '(c)