Mathematics - Application of Derivative Question with Solution | TestHub

We have $f(0)=2$

Now $y-f(a)=f^{\prime }(a)[x-a]$

For x intercept $\mathrm{y}=0$, so

$\mathrm{x}=\mathrm{a}-\frac{\mathrm{f}(\mathrm{a})}{{\mathrm{f}}^{\prime }(\mathrm{a})}=\mathrm{a}-2\Rightarrow \frac{\mathrm{f}(\mathrm{a})}{{\mathrm{f}}^{\prime }(\mathrm{a})}=2$ $\Rightarrow \frac{{\mathrm{f}}^{\prime }(\mathrm{a})}{\mathrm{f}(\mathrm{a})}=\frac{1}{2}$

∴ On integrating both sides w.r.t. a, we get

$\ln \mathrm{f}(\mathrm{a})=\frac{\mathrm{a}}{2}+\mathrm{C}$

$\mathrm{f}(\mathrm{a})={\mathrm{Ce}}^{\mathrm{a}/2}$

$\mathrm{f}(\mathrm{x})={\mathrm{Ce}}^{\mathrm{x}/2}$

$\mathrm{f}(0)=\mathrm{C}\Rightarrow \mathrm{C}=2$

$\therefore \mathrm{f}(\mathrm{x})=2{\mathrm{e}}^{\mathrm{x}/2}$

Hence $\mathrm{k}=2,\mathrm{p}=\frac{1}{2}\Rightarrow \frac{\mathrm{k}}{\mathrm{p}}=4$