Consider the function f(x)=x^{4}−72x^{2}+9, -5\leq x\leq 13. This function has an absolute minimum value =? and an absolute maximum value =?

Consider the function $f\left(x\right)={x}^{4}-72{x}^{2}+9,-5\le x\le 13$.
This function has an absolute minimum value =?
and an absolute maximum value =?
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$f\left(x\right)={x}^{4}-72{x}^{2}+9$
${f}_{prime}\left(x\right)=4{x}^{3}-72\left(2x\right)+0$
${f}_{prime}\left(x\right)=4{x}^{3}-144x$
For the critical numbers ${f}_{prime}\left(x\right)=0$
$4{x}^{3}-144x=0$
$4x\left({x}^{2}-36\right)=0$
$4x\left(x+6\right)\left(x-6\right)=0$
$x=0,-6,6$
Note: $x=-6$ is not in the interval $-5\le x\le 13$
Step 2
Now we find the value of f(x) at the end points and at the critical points
$f\left(x\right)={x}^{4}-72{x}^{2}+9$
$f\left(-5\right)=-1166$
$f\left(0\right)=9$
$f\left(6\right)=-1287$(min)
$f\left(13\right)=16402$(max)
This function has an absolute minimum value = -1287
and an absolute maximum value =16402

Jeffrey Jordon