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

1 Answer
Oct 1, 2014

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.