If f(x) is differentiable and int_0^t^2 x f(x) dx = 2/5t^5, then f (4/25 ) equals

If f(x) is differentiable and int_0^t^2 x f(x) dx = 2/5t^5,… — JEE Main 2004 Mathematics PYQ

AIEEE 2004

f(x)

is differentiable and

\int_0^{t^2} x\, f(x)\, dx = \frac{2}{5}t^5

, then

f\left(\frac{4}{25}\right)

equals

A.
$\frac{2}{5}$
B.
$-\frac{5}{2}$
C.
$1$
D.
$\frac{5}{2}$

Show answer and solution

Correct option: A

Solution

Differentiate both sides with respect to t. By Leibniz rule, the left side gives t² · f(t²) · 2t = 2t³ f(t²). The right side gives 2t⁴. So 2t³ f(t²) = 2t⁴, giving f(t²) = t. Setting t² = 4/25 means t = 2/5, so f(4/25) = 2/5.

[AIEEE 2004]

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