If f and g are differentiable functions in (0, 1)… — JEE Main 2014 Mathematics PYQ

JEE Main 2014

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]

A.
$2f'(c) = g'(c)$
B.
$2f'(c) = 3g'(c)$
C.
$f'(c) = g'(c)$
D.
$f'(c) = 2g'(c)$

Show answer and solution

Correct option: D

Solution

Apply MVT to h(x) = f(x) - 2g(x) on [0,1].

h(0) = f(0) - 2g(0) = 2 - 0 = 2
h(1) = f(1) - 2g(1) = 6 - 4 = 2

Since h(0) = h(1), by Rolle's theorem there exists c ∈ (0,1) such that h'(c) = 0, i.e., f'(c) - 2g'(c) = 0, giving f'(c) = 2g'(c).

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