Relative Minimum and Relative Maximum

The points on the graph where a function changes its increasing, decreasing, or constant behavior help to determine the relative maximum or relative minimum values of the function.

Relative Minimum 

Definition

A function value f(a) is called a relative minimum of f if there exists an interval ( x₁ , x₂ ) that contains a such that

 x₁ < x < x₂  implies f(a) ≤ f(x).

Similarly, when a function changes from decreasing to increasing at a point (c, f(c)), then f is said to have a relative minimum at x=c . The relative minimum value is f(c). 

Relative Maximum

Definition

A function value f(a) is called the relative maximum of f if there exists an interval ( x₁ , x₂ ) that contains a such that 

 x₁ > x > x₂  implies f(a) ≥ f(x).

When a function changes from increasing to decreasing at a point (c,f(c)) , then f is said to have a relative maximum at x=c. The relative maximum value is f(c).

The point marked as the relative maximum is the largest value of the function, while the point marked as the minimum is the smallest value of the function.

Approximating Relative Minimum's and Maximum's 

ex. Use a graphing calculator to approximate the relative minimum and relative maximum of the function f(x) = -x³ + x

After you plug it into your calculator, you can use the zoom and trace buttons in order to estimate the relative minimum and relative maximum.

Relative min: (-0.58, -0.38)

Relative max: (0.58, 0.38)

Application

Problem 
During the 1990's, the sales of bowling equipment in the United States increased and then decreased according to the model

C = 0.165t³ – 7.16t² + 100.6t – 303.1,         8 ≤ t ≤ 16

where C is the sales of the bowling equipment (millions) and t represents  the year, with t=8 corresponding to 1988. Approximate the maximum amount of bowling equipment sales between 1988 and 1996. 

Solution 

Start by graphing the function. After graphing the function, estimate the maximum point on the graph.

Maximum= approximately $158

Video

Kahn Academy- Relative Minimum and Maximum

Works Cited:

Aufmann, Richard N., Vernon C. Barker, and Richard Nation. Precalculus With Limits: A Graphing Approach. Boston: Houghton Mifflin, 2000. Print.

"Identifying Relative Minimum and Maximum Values." Khan Academy. N.p., n.d. Web. 14 Dec. 2014.

"Redirect Notice." Redirect Notice. N.p., n.d. Web. 14 Dec. 2014.