Graphs Of Functions Relative Minima and Maxima A

Graphs Of Functions: Relative Minima and Maxima A relative maximum of a function is a y-value of a point that is greater than the y-values of all the neighboring points in a "local region". The function shown graphed has a relative maximum 4. 4. It occurs at an x-value of - 0. 3. 6 4 4. 4 2 -1 -0. 5 0 0. 5 1 1. 5 -2 The "local region" is within the rectangle. Note the point shown has a y-value > the y-values of all the neighboring points within this rectangle. 2

Graphs Of Functions: Relative Minima and Maxima Example 1: Approximate the relative maximum of the function, f (x) = - x 3 + x 2 + 3 x – 2. Graph the function in the decimal viewing window by pressing Y = , entering the function and pressing 4 4. Next press 2 nd TRACE ZOOM to start the built-in relative maximum location program. "Left Bound? " will appear. Slide 2

Graphs Of Functions: Relative Minima and Maxima Next press eleven times to place the tracer star near but to the left of where the relative maximum occurs. Press ENTER and "Right Bound? " will appear. Next press six times to place the tracer star near but to the right of where the relative maximum occurs. Press ENTER and ”Guess? " will appear. Next, press three times to place the tracer star near the relative maximum. Press ENTER. The relative maximum is 1. 416. Slide 3

Graphs Of Functions: Relative Minima and Maxima A relative minimum of a function is a y-value of a point that is less than the y-values of all the neighboring points in a "local region". The function shown graphed has a relative minimum 2. 2. It occurs at an x-value of 0. 95. 6 4 2. 2 2 -1 -0. 5 0 0. 5 1 1. 5 2 -2 The "local region" is within the rectangle. Note the point shown has a y-value < the y-values of all the neighboring points within this rectangle. Slide 4

Graphs Of Functions: Relative Minima and Maxima A relative minimum can be approximated in a similar fashion as a relative maximum described on slides 2 and 3. The major difference occurs after entering the function where 2 nd TRACE 3 is pressed (instead of 4 to start the built-in relative 2 nd TRACE minimum location program. Try: Approximate the relative maximum of the function, f (x) = 2 x 3/2 – x 2 to three decimal places. The relative maximum is approximately 1. 688. Slide 5

Graphs Of Functions: Relative Minima and Maxima