Let f be a differentiable function such that f(1) = 2 and f'(x) = f(x) for all x ∈ R. If h(x) = f(f(x)), then h'(1) is equal to

Let f be a differentiable function such that f(1) = 2 and… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 12 Jan · Shift II

Let

f

be a differentiable function such that

f(1) = 2

and

f'(x) = f(x)

for all

x \in R

. If

h(x) = f(f(x))

, then

h'(1)

is equal to

A.
$4e^2$
B.
$4e$
C.
$2e$
D.
$2e^2$

Show answer and solution

Correct option: B

Solution

Since f'(x) = f(x), f(x) = Ce^x. Using f(1) = 2 gives C = 2/e, so f(x) = 2e^(x-1).

By chain rule, h'(x) = f'(f(x))·f'(x) = f(f(x))·f(x).

At x = 1: f(1) = 2, so h'(1) = f(f(1))·f(1) = f(2)·2 = 2e^(2-1)·2 = 2e·2 = 4e.

[JEE Main 2019 · 12 Jan · Shift II]

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JEE Main 2019 Mathematics question on Differential Equations