Let $f:(0, \infty) \in R$ and $F(x)=\int\limits_0^x f(t) d t$. If $F\left(x^2\right)=x^2(1+x)$, then $f(4)$ equals
Answer & explanation
Correct answer: option 3
We have,
$F(x)=\int\limits_0^x f(t) d t \Rightarrow F'(x)=f(x) \Rightarrow f(4)=F'(4)$ .......(i)
Now,
$F\left(x^2\right)=x^2(1+x)=x^3+x^2$
$\Rightarrow F'\left(x^2\right) . 2 x=3 x^2+2 x$
$\Rightarrow F'\left(x^2\right)=\frac{3 x+2}{2}$
$\Rightarrow F'(4)=\frac{3 \times 2+2}{2}=4 \Rightarrow f(4)=4$ [Using (i)]