Download Relative Extrema of Functions

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.
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2
Graphs Of Functions: Relative Minima and Maxima
Example 1: Approximate the relative maximum of the
function, f (x) = - x3 + x2 + 3x – 2.
Graph the function in the decimal viewing window by
pressing Y = , entering the function and pressing
4 . Next press 2nd TRACE
4
ZOOM
to start the built-in relative
maximum location program.
"Left Bound?" will appear.
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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.
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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.
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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 2nd TRACE
3 is pressed (instead of
2nd
TRACE
4
to start the built-in relative
minimum location program.
Try: Approximate the relative maximum of the function,
f (x) = 2x3/2 – x2 to three decimal places.
The relative maximum is approximately 1.688.
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Slide 5
Graphs Of Functions: Relative Minima and Maxima
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