If $f\left( x \right)$ is differentiable and $\mathop \smallint \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}$, then $f\left( {\frac{4}{{25}}} \right)$ equals

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1

f\left( x \right)

is differentiable and

\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}

, then

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

equals

View answer and explanation

Correct option: A

\frac{2}{5}

Option A (2/5) is correct because after differentiating both sides of the given equation using the Newton-Leibniz rule and solving for f(x), we find f(4/25) = 2/5.

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