Question

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

Answer 1

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.