# 3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:(vii) $g (x) = \frac{1}{x^2 + 2}$

$g (x) = \frac{1}{x^2 + 2}\\ g^{'}(x) = \frac{-2x}{(x^2+2)^2}\\ g^{'}(x) = 0\\ \frac{-2x}{(x^2+2)^2} = 0\\ x = 0$
$g^{''}(x) = -\frac{-2(x^2+2)^2-(-2x){2(x^2+2)(2x)}}{((x^2+2)^2)^2} \\ g^{''}(0) = \frac{-2\times4}{(2)^4} = \frac{-8}{16} = -\frac{1}{2}< 0$
$g (0) = \frac{1}{0^2 + 2} = \frac{1}{2}$