Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). f(x)=x^{3}-6x^{2} text{on} [-1,5]

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). $f\left(x\right)={x}^{3}-6{x}^{2}\text{on}\left[-1,5\right]$
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Step 1 Let us find the critical points , absolute maximum and minimum of given function Given: $f\left(x\right)={x}^{3}-6{x}^{2}\text{on}\left[-1,5\right]$

Step 2 $f\left(x\right)={x}^{3}-6{x}^{2}$
${f}^{\prime }\left(x\right)=3{x}^{2}-12x$
$0=3x\left(x-4\right)$
$3x=0,x-4=0$
$x=0,x=4$ Critical point $=0,4$

Step 3 To find absolute maximum and absolute minimum: $f\left(x\right)={x}^{3}-6{x}^{2}$
$x=-1$
$f\left(-1\right)={\left(-1\right)}^{3}-6{\left(-1\right)}^{2}$
$f\left(-1\right)=-1-6$
$f\left(-1\right)=-7$
$x=0$
$f\left(0\right)=0-6\left(0\right)$
$f\left(0\right)=0$
$x=4$
$f\left(4\right)={4}^{3}-6{\left(4\right)}^{2}$
$f\left(4\right)=64-6\left(16\right)$
$f\left(4\right)=64-96$
$f\left(4\right)=-32$
$x=5$
$f\left(5\right)={5}^{3}-6{\left(5\right)}^{2}$
$f\left(5\right)=125-150$
$f\left(5\right)=-25$ Absolute maximum at y=0, x=0 Absolute minimum at y=-32, x=5

Step 4 Answer: Critical points at $x=0,4$ Absolute maximum at $x=0$ Absolute minimum at $x=5$