Maximums and Minimums

1. Maximums and Minimums October 6, 2010

2. Objective • SWBAT find the max or min of a function using graphing calculator or algebra

3. Minimum and Maximums • Enter the equation in your calculator • Look at the graph • Decide if you are finding a max or min • If you can’t see a max or a min, change your window • Press y r and choose option 3:minimum or 4:maximum • Use the left and right arrow keys to determine the range the calculator should examine

4. Homework Tip • If you don’t have a graphing calculator, the minimum or maximum can be computed using the following formula: • As long as the equation is of the form: y = ax2 + bx + c

5. Example • f(x) = -x2 + 6x – 8  

6. Example 2. y = -2(x – 1)2 + 2

7. Example • A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45o with respect to the ground. The path of the baseball is given by the function f(x) = -0.0032x2 + x + 3 where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

8. Practice • f(x) = -(x2 + 2x – 3) • f(x) = -x2 – x + 30 • g(x) = x2 + 8x + 11 • f(x) = x2 + 10x + 14 • f(x) = 2x2 – 16x + 31 • f(x) = -4x2 + 24x – 41 • g(x) = 0.5(x2 + 4x – 2)

9. Practice • Find the number of units sold that yields a maximum annual revenue for a sporting goods manufacturer. The total revenue R (in dollars) is given by R = 100x – 0.0002x2, where x is the number of units sold.

10. Practice • A textile manufacturer has daily production costs of C = 100000 – 110x + 0.045x2, where C is the total cost (in dollars) and x is the number of units produced. How many units should be produced each day to yield a minimum cost?

11. Practice • The path of a diver is where y is the height (in feet) and x is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?