Finding Minimum and Maximum Extrema

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Definition of Extrema
Let f be defined on an interval I containing c. f(c) is the minimum of f on I if f(c) < f(c) for all x in I
f(c) is the minimum of f on I if f(c) > f(c) for all x in I
The minimum and max of a function on an interval are the extreme values or extrema of the function on the interval. The minimum and max of a function on an interval are also called the absolute min and absolute max on the interval, respectively.

Definition of Relative Extrema
If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f.
If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f.

Definition of Critical Number
Let f be defined at c. If f '(c) = 0 or if f ' is undefined at c, then c is a critical number of f.

One of the skills that a Calc student must know is how to find Extrema on a Closed Interval. Notice the table of steps below.

Finding Extrema on a Closed Interval

1.	Find the critical numbers of f in (a, b).
2.	Evaluate f at each critical number in (a, b).
3.	Evaluate f at each endpoint of [a, b].
4.	The least of these values is the min and the greatest is the max.

Example:

Find the extrema of f(x) = 3x³ - 4x⁴ on the interval [-1, 2].

Find the critical numbers
f(x) = 3x³ - 4x⁴   {Problem}
f '(x) = 9x² - 16x³   {Differentiate}
9x² - 16x³ = 0   {Set f '(x) = 0}
x(9x - 16) = 0   {Factor}
x = 0, 16/9   {solve to get the critical numbers}

Evaluate f at each critical number in (a,b)
See table below

Evaluate f at each endpoint of [a, b]
See table below

Locate the min and max from the results
See table below

Left Endpoint	Critical Number	Critical Number	Right Endpoint
f(-1) = -7	f(0) = 0	f(16/9) = -23.1	f(2) = -40
	Maximum		Minimum

Another method of finding the extrema of a function is the First Derivative Test.

The First Derivative Test
Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.

If f '(x) changes from negative to positive at c, then f(c) is a relative minimum of f.
If f '(x) changes from positive to negative at c, then f(c) is a relative maximum of f.

An example of this can be seen by clicking here to goto the Increasing or Decreasing Intervals. Note that you are looking at the Sign row for the change of negative to positive or positive to negative. The Min in that example is 2/3 and the Max is 0.

The final method to find the extrema is the Second Derivative Test.

Second Derivative Test
Let f be a function such that f '(c) = 0 and the second derivative of f exists on an open interval containing c.
If f ''(c) > 0, then f(c) is a relative minimum.
If f ''(c) < 0, then f(c) is a relative maximum.
If f ''(c) = 0, then the test fails. In such cases, you can use the First Derivative Test.


Example of the Second Derivative Test

Find the relative extrema for f(x) = -3x⁵ + 5x³

Since the theorem states the f '(c) must = 0, then c must be a critical number. So we get the critical numbers as
x = -1, 0, 1
Next we plug in the x values in f to get the y values which give us the coordinate of the points as
(-1, -2) (0,0) (1, 2)
Now we use the Second Derivative Test to determine the extrema.

Using the Second Derivative Test

Point	Sign of f ''	Conclusion
(-1, -2)	f ''(-1) = 30 > 0	Relative minimum
(0, 0)	f ''(0) = 0 = 0	Test Fails
(1, 2)	f ''(1) = -30 < 0	Relative max

Normally you would use the First Derivative Test on 0, but it isn't an extreme point so don't bother.