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UPSC NDA II 2024 Mathematics Paper

Question Numbers: 95-96

Direction : Consider the following for the items that follow :

Let

f : (-1, 1) \to R

be a differentiable function with

f(0) = -1

and

f'(0) = 1

Let

h(x) = f(2f(x) + 2)

and

g(x) = (h(x))^{2}

Question : 96 of 120

Marks: +1, -0

What is g’(0) equal to?

Solution:

Concept:

This problem uses the chain rule to differentiate a composite function.

Explanation:

Given

f(0) = -1

and

f'(0) = 1

Define

h(x) = f(2f(x) + 2)

and

g(x) = (h(x))^2

Differentiate

g(x)

g'(x) = 2h(x) \cdot h'(x)

g'(0) = 2h(0) \cdot h'(0)

Compute

h(0) = f(2f(0) + 2) = f(2(-1) + 2) = f(0) = -1

Differentiate

h(x)

h'(x) = f'(2f(x) + 2) \cdot 2f'(x)

Thus

h'(0) = f'(2f(0) + 2) \cdot 2f'(0) = f'(0) \cdot 2 \cdot 1 = 1 \cdot 2 = 2

Finally,

g'(0) = 2(-1)(2) = -4

Answer:

-4

(Option A)

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