Question 3: How do you find the relative extrema of a function?

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1 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.

2 First Derivative Test Let f be a non-constant function that is defined at a critical value x c. If f changes from positive to negative at x c, then a relative maximum occurs at the critical point c, f ( c ). If f changes from negative to positive at x c, then a relative minimum occurs at the critical point c, f ( c ). If f does not change sign at x c, then there is no relative extrema at the corresponding critical point.

3 Example 6 Find the Relative Extrema of a Function Find the location of the relative extrema of the function f x x x x 3 ( ) 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 d d d d f x x x x dx dx dx dx x x 3 ( ) So the derivative is ( ) 4 8. f x x x Use sum / difference rule and the constant rule. Use the power rule for derivatives and the fact that the derivative of a constant is zero

4 We need to use this derivative to find the critical values. Set the derivative, ( ) 4 8, equal to zero to find those values. f x x x x x x x 6 30 x0 x30 x x 3 x x x 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 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 never undefined.

5 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: x x 4 x8 0 a b c Identify a, b, and c for the quadratic b b 4ac formula x and put in a the values , Simplify the numerator and denominator

6 To find the critical values with more complicated derivatives, we may need to solve the equation using a graph. The solution to the equation x 4x 8 0 can also be found by locating the x intercepts on the derivative f ( x). f ( x) Figure 5 - The critical values are located at the x intercepts of the derivative, x and x 3.

7 Like factoring or the quadratic formula, the critical values are located at x and x 3. 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 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 vlaues in hand, label them on a number line so that we are able to apply the first derivative test. = 0 = 0 3 f( x) 6 x x3

8 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. (+)(-)(-) = + = 0 (+)(+)(-) = - = 0 (+)(+)(+) = + increasing decreasing 3 increasing f( x) 6 x x3 When 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 x and the relative minimum is located at x 3.

9 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 Relative Maximum: f Relative Minimum: f The relative maximum is located at located at 3,. 37 4, and the relative minimum is