Question

Find the critical points for the curve $f\left(x\right)=\frac{x^4}{4}-\frac{4x^3}{3}+\frac{5x^2}{2}-2x$

Answer 1

Let's take the given curve,
$f\left(x\right)=\frac{x^4}{4}-\frac{4x^3}{3}+\frac{5x^2}{2}-2x$
Critical points are given by,
$f^{\prime }\left(x\right)=0$
$\therefore x^3-4x^2+5x-2=0$
$x=1$ satiesfies the above equation.
$\Rightarrow (x-1)$ is a factor.
On dividing $x^3-4x^2+5x-2$ by $x-1$,
we get, $x^2-3x+2$
$\therefore x^3-4x^2+5x-2=(x-1)(x^2-3x+2)$
$\Rightarrow (x-1)(x^2-3x+2)=0\Rightarrow (x-1)(x-1)(x-2)=0$
$\Rightarrow (x-1{)}^2(x-2)=0\Rightarrow x=1,2$

$\therefore$ Critical points are $1$ and $2$.