Method of DifferentiationmediumFree

Method of Differentiation — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

Let

f

and

g

be two real-valued differentiable functions on

\mathbb{R}

. If

f'(x) = g(x)

and

g'(x) = f(x)

for all

x \in \mathbb{R}

, and

f(3) = 5

g(3) = 4

, then the value of

f^2(\pi) - g^2(\pi)

Solution

Answer: 9

Step 1: Show the expression is constant Let

\phi(x) = f^2(x) - g^2(x)

. Then:

\phi'(x) = 2f(x)f'(x) - 2g(x)g'(x) = 2f(x)g(x) - 2g(x)f(x) = 0

f^2(x) - g^2(x) = k

for all

x \in \mathbb{R}

. Step 2: Determine

k

k = f^2(3) - g^2(3) = 25 - 16 = 9

Hence

f^2(\pi) - g^2(\pi) = 9

. Answer: 9

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