Mathematics - Application of Derivative Question with Solution | TestHub

$f(x)={\int }_0^x{e}^{t^2}(t-2)(t-3)dt$ $\Rightarrow f^{\prime }(x)=e^{x^2}\cdot (x-2)(x-3)$ Put $f^{\prime }(x)=0\Rightarrow x=2,3$ $f^{\prime \prime }(x)=e^{x^2}\cdot 2x\left(x^2-5x+6\right)+e^{x^2}(2x-5)$ $f^{\prime \prime }(2)=-$ ve and $f^{\prime \prime }(3)=+$ ve $\therefore x=2$ is a point of local maxima. and $x=3$ is a point of local minima. Also for $x\in (2,3),f^{\prime }(x)<0$ $\Rightarrow f$ is decreasing on $(2,3)$. Also we observe $f^{\prime \prime }(0)<0$ and $f^{\prime \prime }(1)>0$ $\therefore$ There exists some $C\in (0,1)$ such that $f^{\prime \prime }(C)=0$ Hence all the options are correct.