Question

Find the local maxima and local minima, if any of the given functions. Find also the local maximum and local minimum values, as the case may be:

$g\left(x\right)=\frac{1}{x^2+2}$

Answer 1

Given, $g\left(x\right)=\frac{1}{x^2+2}$

$g^{{}^{\prime }}\left(x\right)=\frac{-2x}{{(x^2+2)}^2}$

Putting this equal to zero, we get
$g^{{}^{\prime }}\left(x\right)=\frac{-2x}{{(x^2+2)}^2}=0$
$\Rightarrow x=0$.

Now let's see the double derivative of this function.
$f^{{}^{\prime \prime }}\left(x\right)=\frac{-1}{2}$
which is negative so function will take maximum value at $x=0$

Maximum value of the function is

$f\left(0\right)=\frac{1}{0+2}=\frac{1}{2}$