Lesson 3-6

Example 1
Locate the extrema for the graph of y = f(x). Name and classify the extrema of the function.

The function has a relative maximum at (-2, 4).	

The function has a relative minimum at (0, 0).

The function appears to have no absolute maximum
or minimum values, since the graph indicates that
the function decreases without bound as x  - and
increases without bound as x  .

Example 2
Use a graphing calculator to graph f(x) = 2x3 - 14x2 + x + 15 and to determine and classify the extrema.

Use a graphing calculator to graph the function in the standard viewing window. Notice that the x-intercepts of the graph are between -1 and 0, 1 and 2, and 6 and 7. Relative maxima and minima will occur somewhere between pairs of x-intercepts.

For a better view of the graph of the function, we need to change the window to encompass the observed x-intercepts more closely.
	
One way to do this is to change the x-axis view to -5 ≤ x ≥ 10. Since the top and bottom of the graph are not visible, you will probably want to change the y-axis view as well. The graph shows -120 ≤ y ≤ 30. From the graph, we can see there is a relative maximum in the interval -1 ≤ x ≤ 1 and a relative minimum in the interval 4 ≤ x ≤ 5.

There are several methods you can use to locate these extrema more accurately.

Method 1:  Use a table of values to locate the approximate greatest and least value of the function. (Hint: Revise TBLSET to begin at -0.5 in intervals of 0.1.)

	
	
There seems to be a relative maximum of 15 at x = 0 and a relative minimum of approximately
-81.97 at x = 4.6.
	
You can adjust the TBLSET increments to hundredths to more closely estimate the
x-value for the relative minimum. A relative minimum of about -81.981 appears to occur at about 4.63.
		
Method 2:  Use the TRACE function to approximate the relative maximum and minimum.

	
	
There seems to be a maximum at x  -0.05 and a minimum at x  4.57.

Method 3:  Use 3:minimum and 4:maximum options on the CALC menu to locate the approximate relative maximum and minimum.

	
	
The calculator indicates a relative maximum of about 15.02 at x = 0.036 and a relative minimum of -81.981 at x = 4.631.

All of these approaches give approximations; some more accurate than others. From these methods, we could estimate that a relative maximum occurs near the point at (0, 15) and a relative maximum near the point at (4.6, -81.9).	

Example 3
The function f(x) = 2x4 - 4x3 has critical points at x = 0 and at x = 1.5. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.

Evaluate the function at each point. Then check the values of the function around each point. Let h = 0.1

x	x - 0.1	x + 0.1		f(x - 0.1)	f(x)	f(x + 0.1)	Type of Critical Point	
0	-0.1	0.1		0.0042	0	-0.0038	inflection point	
1.5	1.4	1.6		-3.2928	-3.375	-3.2768	minimum	

You can verify this solution by graphing f(x) on a graphing calculator.

Example 4
BUSINESS  A company that produces and sells a product has weekly profits, in dollars, that are modeled by the function P(x) = -2x2 + 80x - 40 where x is the number of units of the product sold . The model has a critical point at x = 20.

a.	Determine if this critical point is a maximum.

b.	What does this critical point suggest to the owner about business operations?

a.	Test values around the point. Let h = 0.1.

x	x - 0.1	x + 0.1		f(x - 0.1)	f(x)	f(x + 0.1)	Type of Critical Point	
20	19.9	20.1		759.98	760	759.98	maximum	

	The profit will be at a maximum when the company produces and sells 20 units in a week.

b.	The company’s profits will begin to fall if it tries to produce and sell more than 20 units in a week.