Mathematics - Application of Derivative Question with Solution | TestHub

Given, $F\left(x\right)={\int }_1^x{t}^{2}g\left(t\right)dt$

By Leibnitz rule we get, 

$F^{\text{'}}\left(x\right)=x^{2}g\left(x\right)$ 

$\Rightarrow F^{\text{'}}\left(1\right)=1.g\left(1\right)=0$ $\left(\because g\left(1\right)=0\right)$

Now $F^{\text{'}\text{'}}\left(x\right)=2xg\left(x\right)+x^2{g}^{\text{'}}\left(x\right)$

$\Rightarrow F^{\text{'}\text{'}}\left(x\right)=2xg\left(x\right)+x^{2}f\left(x\right)$ $\left(\because g^{\text{'}}\left(x\right)=f\left(x\right)\right)$

$\Rightarrow F^{\text{'}\text{'}}\left(1\right)=0+1\times 3$

$\Rightarrow F^{\text{'}\text{'}}\left(1\right)=3$

$F\left(x\right)$ has a local minimum at $x=1$.