# How do you find the absolute extreme values of a function on an interval?

Oct 1, 2014

How to Find Absolute Extrema of a Function on $\left[a , b\right]$

Step 1: Find all critical values of $f$ on $\left(a , b\right)$.
Step 2: Evaluate $f$ at the critical values from Step 1 and at the endpoints a and b.
Step 3: Choose the largest value as the absolute maximum value,
and choose the smallest value as the absolute minimum value.

Let us find the absolute extrema of $f \left(x\right) = {x}^{3} - 6 {x}^{2} + 9 x$ on $\left[- 1 , 2\right]$.

Step 1

$f ' \left(x\right) = 3 {x}^{2} - 12 x + 9 = 3 \left(x - 1\right) \left(x - 3\right) = 0$

$R i g h t a r r o w x = 1 , 3$, but only $x = 1$ is on $\left(- 1 , 2\right)$.

Step 2

$f \left(- 1\right) = {\left(- 1\right)}^{3} - 6 {\left(- 1\right)}^{2} + 9 \left(- 1\right) = - 16$

$f \left(1\right) = {\left(1\right)}^{3} - 6 {\left(1\right)}^{2} + 9 \left(1\right) = 4$

$f \left(2\right) = {\left(2\right)}^{3} - 6 {\left(2\right)}^{2} + 9 \left(2\right) = 2$

Step 3

Hence,

{("Absolute Maximum: " f(1)=4), ("Absolute Minimum: " f(-1)=-16):}

I hope that this was helpful.