Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{l}{\varphi }_3(\mathrm{m})=3+2\mathrm{m}-7{\mathrm{m}}^2+2{\mathrm{m}}^3\\ {\varphi }_2(\mathrm{m})=-14\mathrm{m}+7{\mathrm{m}}^2\\ {\varphi }_3^{\prime }(\mathrm{m})=2-14\mathrm{m}+6{\mathrm{m}}^2\end{array}$ Now, putting ${\varphi }_3(\mathrm{m})=0$, we have $3+2m-7m^2+2m^3=0$ $\begin{array}{l}\Rightarrow (1-\mathrm{m})(1+2\mathrm{m})(3-\mathrm{m})=0\\ \Rightarrow \mathrm{m}=-\frac{1}{2},1,3\end{array}$ We know that $c{\varphi }_n^{\prime }(m)+{\varphi }_{n-1}(m)=0$, which in the given case becomes $\begin{array}{rl}& c\left(2-14m+6m^2\right)+\left(-14m+7m^2\right)=0\\ \Rightarrow & c=\frac{14m-7m^2}{2-14m+6m^2}\end{array}$ So, when $\mathrm{m}=-\frac{1}{2},\mathrm{c}=-\frac{5}{6}$ When $\mathrm{m}=1,\mathrm{c}=-\frac{7}{6}$ When $\mathrm{m}=3,\mathrm{c}=-\frac{3}{2}$ $\therefore$ Asymptotes are $\mathrm{y}=-\frac{1}{2}\mathrm{x}-\frac{5}{6}$, $y=x-\frac{7}{6}$ and $y=3x-\frac{3}{2}$