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) = x^3 – 3x

Given, $f(x)=x^{2}-3 x$
Differentiate with respect to $x$ then we get,
$
f(x)=3 x^{2}-3
$
$\mathrm{Now}, \mathrm{f}(x)=0$
$
3 x^{2}=3 \Rightarrow x=\pm 1
$
Again differentiate $f(x)=3 x^{2}-3$
$
\begin{array}{l}
f^{\prime}(x)=b x \\
f^{\prime}(1)=6>0 \\
f^{\prime}(-1)=-6<0
\end{array}
$
By second derivative test, $x=1$ is a point of local minima and local minimum value off at $x=1$ is $f(1)=1^{2}-3=1-3=-2$
However, $x=-1$ is a point of local maxima and local maxima value offf at $x=-1$ is
$
\begin{array}{l}
f(-1)=(-1)^{3}-3(-1) \\
=-1+3 \\
=2
\end{array}
$
Hence, the value of minima is $-2$ and maxima is 2 .