QJEEAdv12Ma2193: JEE Advanced 2012 Mathematics - Class 12 JEEAdv12Ma2193 Solution: If for all then

$\begin{aligned}& \text { Given, } f(x)=\int_0^x e^{t^2}(t-2)(t-3) d t, x \in(0, \infty) \\& \Rightarrow \quad f^{\prime}(x)=e^{x^2}(x-2)(x-3)\end{aligned}$

Here, $f^{\prime}(x)$ changes its sign $+v e$ to $-v e$ about $x=2$ and $f^{\prime}(x)$ changes its sign $-v \mathrm{e}$ to $+v \mathrm{e}$ about $x=3$

Hence, $x=2$ is the point of local maxima and $x=3$ is the point of local minima

\because f^{\prime}(x)<0 for $x \in(2,3)$

$\therefore f(x)$ is decreasing on $x \in(2,3)$

$\because \quad f^{\prime}(x)$ is continuous and differentiable for all $x(0, \infty)$ and $f^{\prime}(2)=f^{\prime}(3)=0$

$\therefore$ According to Rolle's theorem, $f^{\prime \prime}(c)=0$ must have at least one root $\in(2,3)$