Transcript Properties of Real Numbers

Relative Maxima and Minima
Eric Hoffman
Calculus
PLHS
Nov. 2007
Key Topics
• Critical Numbers: the x-values at which the f ‘(x)=0
or f ‘(x) fails to exist
• Note: The critical numbers are the points where the
graph will switch from increasing to decreasing or
vice versa
• Find the critical numbers for the following functions:
f(x) = 3x2 – 6x + 3
x=1
f(x) = x3/2 – 3x + 7
x=4
Key Topics
• Relative maximum: the highest value for f(x) at that
particular “peak” in the graph
• Relative minimum: the lowest value for f(x) at that
particular “valley” in the graph
Relative maximum
Relative maximum
Relative minimum
Relative minimum
Relative minimum
Key Topics
• How to determine whether it is a relative maximum or
a relative minimum at a focal point:
Step 1: Find the focal points of the graph to determine the
intervals on which f(x) is increasing or decreasing
Step 2: Choose an x-value in each interval to determine
whether the function is increasing or decreasing within
that interval
Step 3: If f(x) switches from increasing to decreasing at a
focal point, there is a relative maximum at that focal point
If f(x) switches from decreasing to increasing at a focal point,
there is a relative minimum at that focal point
Key Topics
+ to - means maximum
means
minimum
•- to
It +
might
help
to make a number line displaying
your findings
- - - | +++++++ | - - - - | +++++++ | - - - - - - - | +++
• Another helpful method might be to make a
3 – 3x2 – 12x + 1
table of your findings
f(x)
This tells us
that=in2x
this interval
the
Interval
(-∞,-1)
(-1,2)
(2,∞)
function is increasing
us that in this interval the
TestThis
# tells
‘(t)
Sign
function isf decreasing
us that in this interval the
t =This
-2 tells
24
function is increasing
t=0
t=3
-12
24
of f ‘(t)
+
+
Key Topics
• Homework: pg. 186 1 – 22 all