Mathematics - Application of Derivative Question with Solution | TestHub

Let $y=x{e}^x$. Differentiate both side w.r.t. 'x'. $$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{e}}^{\mathrm{x}}+{\mathrm{xe}}^{\mathrm{x}}={\mathrm{e}}^{\mathrm{x}}(1+\mathrm{x})$$ Put $\frac{\mathrm{dy}}{\mathrm{dx}}=0$ $$\begin{array}{l}\Rightarrow {\mathrm{e}}^{\mathrm{x}}(1+\mathrm{x})=0\\ \Rightarrow \mathrm{x}=-1\end{array}$$ Now, $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}={\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{\mathrm{x}}(1+\mathrm{x})={\mathrm{e}}^{\mathrm{x}}(\mathrm{x}+2)$ $${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}_{(\mathrm{x}=-1)}=\frac{1}{\mathrm{e}}+0>0$$ Hence, $\mathrm{y}={\mathrm{xe}}^{\mathrm{x}}$ is minimum function and $${\mathrm{y}}_{\min }=-\frac{1}{\mathrm{e}}\text{ . }$$