Question

Find all the points of local maxima and local minima of the function $f$ given by $f\left(x\right)=2x^3-6x^2+6x+5$.

Answer 1

$f\left(x\right)=2x^3-6x^2+6x+5$
Differentiating w.r.t. $x,$
$f^{\prime }\left(x\right)=6x^2-12x+6$
$\Rightarrow f^{\prime }\left(x\right)=6(x^2-2x+1)$
$\Rightarrow f^{\prime }\left(x\right)=6(x-1{)}^2$
Putting $f^{\prime }\left(x\right)=0$
$6(x-1{)}^2=0$
$\Rightarrow (x-1{)}^2=0$
$\Rightarrow x=1$
Crtical point is $x=1$

$\because f^{\prime }\left(x\right)=6(x-1{)}^2$
Again differentiating w.r.t. $x,$
$\Rightarrow f^{\prime \prime }\left(x\right)=6\times 2(x-1)$
$\Rightarrow f^{\prime \prime }\left(x\right)=12(x-1)$
Putting $x=1$
$f^{\prime \prime }\left(1\right)=0$
As second derivative also fails.
Using first derivative test, we get
We know that the sign of
$f^{\prime }\left(x\right)=6(x-1{)}^2$ will not change around $x=1$
Hence, $x=1$ is neither a points of local maxima nor a point of local minima, it is a point of inflexion.