Extreme Value of Functions For a parabola, the extreme value is the

Extreme Value of Functions For a parabola, the extreme value is the
Extreme Value of Functions
For a parabola, the extreme value is the vertex. If the parabola opens up, then the vertex is a minimum point. If the parabola opens down, then the vertex is a maximum point.
1
Vertex Form of a parabola: y=a(x­h)2 + k (h,k) is the vertex
Example: Express in vertex form.
Sketch the graph.
Find the maximum or minimum value.
f(x) = ­x2 + x + 2
y
5
4
3
2
1
x
­5
­4
­3
­2
­1
0
1
2
3
4
­1
­2
­3
­4
­5
2
Example: Express in vertex form.
Sketch the graph.
Find the maximum or minimum value.
f(x) = 2x2 + 4x + 5
y
5
4
3
2
1
x
­5
­4
­3
­2
­1
0
1
2
3
4
­1
­2
­3
­4
­5
3
Example:
Most cars get their best gas mileage when traveling at a relatively modes speed. The gas mileage, M, for a certain car model is modeled by the function M(s) =
for 15 ≤ s ≤ 70, where s is the speed in mph and M is
measured in mpg.
What is the cars best gas mileage?
4
Local maximum value:
If a point (a,f(a)) is the highest point on the graph of f
within a certain range (not on the edge) then f(a) is the local maximum value.
Local minimum value:
If a point (a,f(a)) is the lowest point on the graph of f within a certain range (not on the edge) then f(a) is the
local minimum value.
See Figure 4 on page 203
5
Use the graphing calculator to find the local max/min values f(x) = x3 ­ 8x + 1
for:
f(x) = x4 ­ 2x3 ­ 11x2
f(x) = x√6-x
f(x) = 1 - x2
x3
6
Assignment
page 204 16­34E, 40
7
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