Question

Find the critical points of the following functions and test them for their maxima and minima.
$f\left(x\right)=\frac{x^2}{17}-\ln \left(x^{2}8\right)$

Answer 1

Given: $f\left(x\right)=\frac{x^2}{17}-\ln \left(x^{2}8\right)$

$f\left(x\right)$ is not defined as $x=0$ since $\ln 0$ is not defined

$f^{\prime }\left(x\right)=\frac{2x}{17}-\frac{1}{x^2.8}\cdot 8.2x=10$

$\Rightarrow \frac{2x}{17}-\frac{2}{x}=0$

$\Rightarrow \frac{x}{17}-\frac{1}{x}=0$

$\Rightarrow x^2-17=0\Rightarrow x=\pm \sqrt{17}$

here $f^{\prime \prime }\left(x\right)=\frac{2}{17}+\frac{2}{x^2}$

at $x=\sqrt{17}$ and $x=-\sqrt{17}$

So $f^{\prime \prime }\left(x\right)>0$ (So minima at $x=\sqrt{17}$ and $-\sqrt{17})$