Let f:(0,∞) → R and F(x) = int_0^x f(t) dt. If F(x^2) = x^2(1+x), then f(4) equals

Let f:(0,∞) → R and F(x) = int_0^x f(t) dt. If F(x^2) =… — JEE Main 2001 Mathematics PYQ

2001

Let

f:(0,\infty) \to R

and

F(x) = \int_0^x f(t)\, dt

. If

F(x^2) = x^2(1+x)

, then

f(4)

equals

A.
$\dfrac{5}{4}$
B.
$7$
C.
$4$
D.
$2$

Show answer and solution

Correct option: C

Solution

Differentiate both sides of F(x²) = x²(1+x) with respect to x:

F'(x²)·2x = 2x(1+x) + x²

So f(x²)·2x = 2x + 2x² + x²= 2x + 3x²

f(x²) = 1 + (3x/2)

To find f(4), set x² = 4, so x = 2:

f(4) = 1 + 3(2)/2 = 1 + 3 = 4.

[2001]

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