Mathematics - Application of Derivative Question with Solution | TestHub

(A) Let $g(t)=e^{\frac{t^2}{2}}\cdot f(t)$ for $t\in [a,b]$ here $g(a)=g(b)=0$

According to Rolle's theorem

${\mathrm{g}}^{\prime }(\mathrm{x})=0$, for some $\mathrm{x}\in (\mathrm{a},\mathrm{b})$

$\Rightarrow x{e}^{\frac{x^2}{2}}\cdot f(x)+e^{\frac{x^2}{2}}\cdot f^{\prime }(x)=0$

$\Rightarrow \mathrm{xf}(\mathrm{x})+{\mathrm{f}}^{\prime }(\mathrm{x})=0$

(B) Let $g(t)=e^{\frac{-t^2}{2}},f(t)$, for $t\in [a,b]$

here $g(a)=g(b)=0$ ∴

According to Rolle's theorem

${\mathrm{g}}^{\prime }(\mathrm{x})=0$ for some $\mathrm{x}\in (\mathrm{a},\mathrm{b})$

$\Rightarrow -x{e}^{\frac{x^2}{2}}\cdot f(x)+e^{\frac{-x^2}{2}}\cdot f(x)=0$

$\Rightarrow \mathrm{xf}(\mathrm{x})-{\mathrm{f}}^{\prime }(\mathrm{x})=0$

(C) Let $g(t)=e^{50t}f(t),t\in [a,b]$

Here $g(a)=g(b)=0$

∴ According to Rolle's theorem

${\mathrm{g}}^{\prime }(\mathrm{x})=0$ for some $\mathrm{x}\in (\mathrm{a},\mathrm{b})$

$\Rightarrow 50e^{50x}\mathrm{f}(\mathrm{x})+{\mathrm{e}}^{50\mathrm{x}}{\mathrm{f}}^{\prime }(\mathrm{x})=0\Rightarrow 50\mathrm{f}(\mathrm{x})+{\mathrm{f}}^{\prime }(\mathrm{x})=0$

(D) Let $g(t)=50^{t}f(t),t\in [a,b]$

Here $g(a)=g(b)=0$

∴ According to Rolle's theorem

${\mathrm{g}}^{\prime }(\mathrm{x})=0$ for some $\mathrm{x}\in (\mathrm{a},\mathrm{b})$

$\Rightarrow 50^x\ln 50\mathrm{f}(\mathrm{x})+50^{\mathrm{x}}{\mathrm{f}}^{\prime }(\mathrm{x})=0$

$\Rightarrow (\ln 50)\mathrm{f}(\mathrm{x})+{\mathrm{f}}^{\prime }(\mathrm{x})=0$