Question

If f(x) is differentiable and $\int_0^{t^2}x f(x) dx =\frac{2}{5}t^5,$ then $f\bigg(\frac{4}{25}\bigg)$ equals

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

Answer 1

The Correct Option is A

Here, $\int_0^{t^2}x f(x) dx =\frac{2}{5}t^5,$
Using Newton Leibnitz's formula, differentiating both
sides, we get
$t^2 \{ f(t^2)\} \bigg \{\frac{d}{dt}(t^2)\bigg\}-0.f(0) \bigg\{\frac{d}{dt}(0)\bigg\}=2t^4$
$\Rightarrow \, \, \, \, t^2f(t^2)2t=2t^4 \, \, \, \Rightarrow \, \, f(t^2)=t$
$\therefore \, \, \, \, \, \, \, f\bigg(\frac{4}{25}\bigg)=-\frac{2}{5} \, \, \, \, \, \, \, \, \, \, \bigg[ putting \, t=\frac{2}{5}\bigg]$
$\Rightarrow \, \, \, \, f\bigg(\frac{4}{25}\bigg)=\frac{2}{5}$