Question

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 $c \in \left] {0,1} \right[.$
A. 2 f’(c) = g’(c)
B. 2 f’(c) = 3g’(c)
C. f’(c) = g’(c)
D. 2 f’(c) = 2g’(c)

Answer 1

Hint : Here, it is given that f and g are differentiable in (0, 1) therefore f and g are also continuous in [0, 1]. We can apply here Rolle’s Theorem to find the correct statement for c. Relate f and g and apply Rolle’s Theorem to get the result.

Complete step-by-step answer :
Here, f(0) = 2 = g(1), g(0) = 0 and f(1) = 6
Since, f and g are differentiable in (1, 0)
Let us assume another function h (x) such that
$h(x) = f(x) – 2g(x)$
$\Rightarrow$ h (0) = f(0) – 2g(0)
Putting values of f(0) and g(0) as given.
⇒ $h (0) = 2 – 0 = 2$
Now, h (1) = f (1) – 2g (1) = 6 – 2 × 2
$H(1) = 2, h(0) = h(1) = 2$
Using Rolle’s Theorem,
There exists $c \in \left] {0,1} \right[,$ such that h’(c) = 0
Rolle’s Theorem: Rolle’s Theorem states that if f(x) is a function, such that f(x) is continuous on the closed interval [a, b] and also differentiable on the open interval (a, b), then there exists a point c in (a, b) that is, a < c < b such that f’(c) = 0.
$\Rightarrow$ f’(c) – 2g’(c) = 0, for some $c \in \left] {0,1} \right[.$
$\Rightarrow$ f’(c) = 2g’(c)

So, the correct answer is “Option D”.

Note : In these types of questions, first check the conditions which are compulsory for Rolle’s Theorem and then apply Rolle’s Theorem. In this question we have to choose a correct relation from the options given, so try to establish a relation as given options.