Question 3: How do you find the relative extrema of a function?
The strategy for tracking the sign of the derivative is useful for more than determining where a function is increasing or decreasing. It is also useful for locating the relative extrema of a function. At a relative extrema, a function changes from increasing to decreasing or decreasing to increasing. The number lines in the previous question allow us to see these changes by observing changes in the sign of the derivative of a
function.
When the derivative of a function changes from positive to negative, we know the
function changes from increasing to decreasing. As long as the function is defined at the critical value where the change occurs, the critical point must be a relative maximum. If the derivative of a function changes from negative to positive, we know the function changes from decreasing to increasing. In this case, the critical point is a relative
minimum as long as the function is defined there. If the derivative does not change sign at a critical value, there is no relative extrema at the corresponding critical point.
The First Derivative Test summarizes these observations and helps us to locate relative extrema on a function.
First Derivative Test
Let f be a non-constant function that is defined at a critical value xc.
If fchanges from positive to negative at xc, then a relative maximum occurs at the critical point
c f c, ( )
. If fchanges from negative to positive at xc, then a relative minimum occurs at the critical point
c f c, ( )
.If f does not change sign at xc, then there is no relative extrema at the corresponding critical point.
Example 6 Find the Relative Extrema of a Function
Find the location of the relative extrema of the function
3 2
( ) 4 21 18 5 f x x x x
Solution The first derivative test requires us to construct a number line for the derivative so that we can identify where the graph is increasing and decreasing. Using the rules for derivatives, the first derivative of the function f x( ) is
3 2 2 ( ) 4 21 18 5 4 3 21 2 18 1 0 d d d d f x x x x dx dx dx dx x x So the derivative is 2 ( ) 12 42 18 f x x x .Use Sum / Difference Rule and the Product with a Constant Rule.
Use the Power Rule for
Derivatives and the fact that the derivative of a constant is zero
We need to use this derivative to find the critical values. Set the derivative, f x( ) 12 x242x18, equal to zero to find those values.
2 2 1 2 12 42 18 0 6 2 7 3 0 6 2 1 3 0 2 1 0 3 0 2 1 3 x x x x x x x x x x x In general, critical values may also come from x values where the derivative is undefined. Since f x( ) is a polynomial, it is defined everywhere so the derivative is defined everywhere.
Although this derivative could be factored to find the critical values, most quadratic derivatives are not factorable. In this case, the quadratic equation yielding the critical values can be solved using the quadratic formula. This strategy would yield the same critical values as factoring:
2 2 1 2 12 42 18 0 42 42 4 12 18 2 12 42 900 24 42 30 24 3, a b c x x x To find the critical values with more complicated derivatives, we may
Set the derivative equal to zero
Factor the greatest common factor from each term
Factor the trinomial
To find where the product is equal to zero, set each factor equal to zero and solve for the variable
Identify a, b, and c for the quadratic formula 2 4 2 b b ac x a and put in the values
2
12x 42x180 can also be found by locating the x intercepts on the derivative f x( ).
Like factoring or the quadratic formula, the critical values are located at
1 2
x and x3. All three strategies yield the same critical values. Keep in mind that a graph will give approximate values while factoring or the quadratic formula yield exact values.
If the derivative is not factorable, linear, or quadratic, another method will need to be used to determine where the derivative is equal to zero. Even though a graph of the derivative only gives an estimate of the critical values, it may be the only way to find the critical values if the derivative is complicated.
With the critical values in hand, label them on a number line so that we are able to apply the first derivative test.
If we select a test point in each interval and determine the sign of each factor, we can complete the number line and track the sign of the derivative.
( ) 6 2 1 3 f x x x = 0 1 2 = 0 3When a continuous function changes from increasing to decreasing, we have a relative maximum at the critical value. When a continuous
function changes from decreasing to increasing, we have a relative minimum at the critical value. In this case, the relative maximum is located at 1
2
x and the relative minimum is located at x3.
To find the ordered pairs for the relative extrema, we need to substitute the critical values into the original function f x( ) to find the
corresponding y values:
3 2 37 1 1 1 1 2 2 2 2 4 3 2 Relative Maximum: 4 21 18 5 Relative Minimum: 3 4 3 21 3 18 3 5 22 f f The relative maximum is located at
1 37
2, 4 and the relative minimum is
located at
3, 22
.( )
f x
(+)(-)(-) = + (+)(+)(-) = - (+)(+)(+) = +
increasing decreasing increasing
( ) 6 2 1 3 f x x x = 0 1 2 = 0 3Example 7 Find the Relative Maximum
Find the location of the relative maximum of the function
( ) xx g x
e
Solution The derivative of g x( )is found with the Quotient Rule for Derivatives with 1 x x u x v e u v e
Put these expressions into the Quotient Rule:
2 1 ( ) x x x e x e g x e To make it easier to find the critical values, simplify the derivative.
2
2
1 1 1 x x x x x x e x e e x e e x e Now we can find the critical values by determining where this fraction is equal to zero or undefined.
Any fraction is equal to zero where the numerator is equal to zero. In this case, this is where 1 x 0 or x1. Fractions are undefined where the denominator is equal to zero. The denominator for this fraction is x
e
and is always positive so there are no x values where the fraction is undefined.
The Quotient Rule is d u vu 2uv
dx v v
Factor ex from the numerator
Reduce the common ex factor in the numerator and denominator
To apply the first derivative test, label a number line with this critical value and test the first derivative on either side of the critical value:
Since this function is continuous and the derivative changes from increasing to decreasing at x1, the critical point is a relative maximum. The y value for the relative maximum comes from the function g x( ) xx
e
and is g(1) 11 e
. The ordered pair for the relative
maximum is
1,1 e .Example 8 Find the Minimum Average Cost
The total daily cost to produce Q units of a product is given by the function
20.005 10 1000 dollars
C Q Q Q
The average cost function C Q
is found by dividing the total daily cost function C Q
by the quantity Q.a. Find the average cost function C Q
.Solution The average cost function is defined by C Q
C Q
Q .
Substitute the total daily cost function into the numerator of this fraction
increasing decreasing 1 ( ) xx f x e = 0 1
0.005Q2 10Q 1000C Q
Q
b. Find the quantity that yields the minimum average cost.
Solution The minimum average cost is found by locating the relative
minimum of the average cost function. To use the first derivative test to find this relative minimum, we need to take the derivative of C Q
using the Quotient Rule for Derivatives. The numerator, denominator and their derivatives are 2 0.005 10 1000 0.010 10 1 u Q Q v Q u Q v The derivative of the average cost function is
2 2 2 0.010 10 0.005 10 1000 1 uv vu v Q Q Q Q C Q Q Since we need to use the derivative to find the critical values, we simplify the derivative as much as possible:
2 2 2 2 2 0.010 10 0.005 10 1000 0.005 1000 Q Q Q Q C Q Q Q Q Any fraction is equal to zero when the numerator is equal to zero and undefined where the denominator is equal to zero.
Simplify the numerator by carrying out the multiplication and
subtraction
2
2
2
Set the numerator equal to 0
0.005 1000 0 0.005 1000 200000 200000 447 Q Q Q Q Q 2
Set the denominator equal to 0 0 0 Q Q
Since quantities produced must be positive, only Q447 is a reasonable critical value for this function.
The number line for this function only includes positive quantities. Testing on either side of the critical numbers yields the behavior of C .
Since the function decreases on the left side of the critical value and increases on the right side of the critical value, the quantity at
approximately 447 units is a relative minimum. The minimum average cost is obtained from
0.005Q2 10Q 1000 C Q Q and is calculated as
decreasing increasing
22 0.005Q 1000 C Q Q = 0 447 0
2 0.005 200000 10 200000 1000 200000 200000 14.47 C Since the total daily cost is in dollars and we are dividing by the number of units to get the average cost, the units on the average cost are
dollars per unit. This means that a production level of about 447 units gives the lowest average cost of 14.47 dollars per unit.
Figure 6 - The relative minimum for the average cost function.