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Find f(4) given F(x²)=x²(1+x), F=∫f | JEE

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

Question

Let

f:(0,\infty)\to\mathbb{R}

and

F(x)=\displaystyle\int_{0}^{x}f(t)\,dt

. If

F\!\big(x^{2}\big)=x^{2}(1+x)

, then

f(4)

equals

\dfrac54

7

4

correct

2

Solution

Step 1: By the Fundamental Theorem of Calculus,

F'(x)=f(x)

. The given relation is

F\!\big(x^{2}\big)=x^{2}+x^{3}.

Step 2: Differentiate both sides with respect to

x

, using the chain rule on the left:

F'\!\big(x^{2}\big)\cdot 2x=2x+3x^{2}.

Step 3: Solve for

F'\!\big(x^{2}\big)=f\!\big(x^{2}\big)

f\!\big(x^{2}\big)=\dfrac{2x+3x^{2}}{2x}=1+\dfrac{3}{2}x.

Step 4: To obtain

f(4)

, set

x^{2}=4

, i.e.

x=2

f(4)=1+\dfrac{3}{2}(2)=1+3=4.

Correct answer: (3)

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