Mathematics - Limits Question with Solution | TestHub

Using L'Hospital Rule$\left(asitis\frac{0}{0}form\right)$
$\underset{x\to 2}{\lim }\frac{f^{\prime }\left(x\right) g\left(x\right)+f\left(x\right) g^{\prime }\left(x\right)}{f^{″}\left(x\right) g^{\prime }\left(x\right)+f^{\prime }\left(x\right) g^{″}\left(x\right)}=1$
$\Rightarrow \frac{f \left(2\right) g^{\prime }\left(2\right)}{f^{″} \left(2\right) g^{\prime }\left(2\right)}=1\Rightarrow f^{″}\left(2\right)=f\left(2\right)>0$
$Also{f}^{\prime }\left(2\right)=0\text{ and }{f}^{″}\left(2\right)>0$
$\therefore$$x=2$is local minima.