Question

Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.

$g\left(x\right)=x^3-3x$

Answer 1

Given function is, $g\left(x\right)=x^3-3x$
$\therefore g^{\prime }\left(x\right)=3x^2-3and{g}^{\prime \prime }\left(x\right)=6x$
For maxima or minima put g'(x)=0
$\therefore 3x^2-3=0\Rightarrow x=\pm 1$
Thus, we expect extremum only at two points -1 and 1
At $x=-1,g"(-1)=6(-1)=-6<0$
$\therefore$ g has a local maxima at x = -1 and local maximum value = g(-1)
$=(-1{)}^3-3(-1)=-1+3=2$
At $x=1,g"\left(1\right)=6\times 1=6>0$
$\therefore$ g has a local minima at x = -1 and local minimum value =g(-1)
$=(1{)}^3-3(1)=1-3=-2$
At $x=1,g"\left(1\right)=6\times 1=6>0$
$\therefore g$ has a local minima at x= 1 and local minimum value = g(1)
$=1^3-3\times 1=-2$