Question

If $f:(0,\infty )\to R$ and $F\left(x\right)={\int }_0^{x}f\left(t\right)$ dt. If $F\left(x^2\right)=x^2(1+x)$ then f(1) =

• A. $\frac{5}{4}$
• B. $\frac{5}{2}$
• C. $4$
• D. $2$

Answer 1

The correct option is B $\frac{5}{2}$

$F\left(x\right)={\int }_0^{x}f\left(t\right)dt\Rightarrow F^{\prime }\left(x\right)=f\left(x\right)\Rightarrow F^{\prime }\left(1\right)=f\left(1\right)$

$F\left(x^2\right)=x^2(1+x)F^{\prime }\left(x^2\right)\cdot 2x=2x(1+x)+x^2\Rightarrow F^{\prime }\left(1\right)=f\left(1\right)=\frac{5}{2}$