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

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

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Answer 1 · 2014-10-01T01:03:29

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

Step 1: Find all critical values of $f$ on $(a,b)$ .
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 $f(x)=x^3-6x^2+9x$ on $[-1,2]$ .

Step 1

$f'(x)=3x^2-12x+9=3(x-1)(x-3)=0$

$Rightarrow x=1,3$ , but only $x=1$ is on $(-1,2)$ .

Step 2

$f(-1)=(-1)^3-6(-1)^2+9(-1)=-16$

$f(1)=(1)^3-6(1)^2+9(1)=4$

$f(2)=(2)^3-6(2)^2+9(2)=2$

Step 3

Hence,

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

I hope that this was helpful.

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

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