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.

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Answer 2

To find the absolute extreme values of a function on an interval, follow these steps:

  1. Determine the critical points of the function within the given interval by finding where the derivative of the function is equal to zero or undefined.
  2. Evaluate the function at each critical point and at the endpoints of the interval.
  3. Compare the values obtained in step 2 to identify the absolute maximum and minimum values of the function on the given interval.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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