Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be: f(x)=1/(x^2+2)

As per the given question,

Therefore

\[x\text{ }=\text{ }0,\]

now for the values close to \[x\text{ }=\text{ }0,\] and to the left of \[0,\text{ }f’\left( x \right)\text{ }>\text{ }0\]

Also for values

\[x\text{ }=\text{ }0,\]

and to the right of \[0,\text{ }f’\left( x \right)\text{ }>\text{ }0\]

Therefore, by first derivative test,\[x\text{ }=\text{ }0,\] is a point of local maxima and local minima value of \[f\text{ }\left( x \right)\text{ }is\text{ }{\scriptscriptstyle 1\!/\!{ }_2}.\]