Mathematics - Application of Derivative Question with Solution | TestHub

Let $m$ be the slope of the tangent to the curve $y=e^x\cos x$ Then, $m=\frac{dy}{dx}=e^x(\cos x-\sin x)$ Diff. w.r.t 'x' $$\begin{array}{r}\Rightarrow \frac{\mathrm{dm}}{\mathrm{dx}}={\mathrm{e}}^{\mathrm{x}}(\cos \mathrm{x}-\sin \mathrm{x})+{\mathrm{e}}^{\mathrm{x}}(-\sin \mathrm{x}-\cos \mathrm{x})\\ =-2{\mathrm{e}}^{\mathrm{x}}\sin \mathrm{x}\end{array}$$ and $\frac{{\mathrm{d}}^2\mathrm{m}}{{\mathrm{dx}}^2}=-2{\mathrm{e}}^{\mathrm{x}}(\sin \mathrm{x}+\cos \mathrm{x})$ Put $\frac{\mathrm{dm}}{\mathrm{dx}}=0\Rightarrow \sin \mathrm{x}=0\Rightarrow \mathrm{x}=0,\pi ,2\pi$ Clearly, $\frac{{\mathrm{d}}^2\mathrm{m}}{{\mathrm{dx}}^2}>0$ for $\mathrm{x}=\pi$ Thus, $y$ is minimum at $x=\pi$. Hence the value of $\alpha =\pi$