Question

Find the points of local maxima and local minima of the function $f$ given by $f\left(x\right)=x^3-3x+3$.

Answer 1

$f\left(x\right)=x^3-3x+3$
Differentiation w.r.t. $x$
$f^{\prime }\left(x\right)=3x^2-3+0$
$\Rightarrow f^{\prime }\left(x\right)=3(x^2-1)$
Putting $f^{\prime }\left(x\right)=0$
$\Rightarrow 3(x^2-1)=0$
$\Rightarrow x^2-1=0$
$\Rightarrow (x-1)(x+1)=0$
$\Rightarrow x=-1,1$
Critical points are $x=-1,1$
Sign scheme of $f^{\prime }\left(x\right):$

Finding local maxima and minima for
$f\left(x\right)=x^3-3x+3$
Local minima: $f\left(1\right)=1^3-3\left(1\right)+3=1$
Local maxima:
$f(-1)=(-1{)}^3-3(-1)+3=5$