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 f(x) = x3/2 – 3x + 7

x = 1 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

• It might help to make a number line displaying your findings

- - - | +++++++ | - - - - | +++++++ | - - - - - - - | +++

- to + means minimum+ to - means maximum

• Another helpful method might be to make a table of your findings f(x) = 2x3 – 3x2 – 12x + 1

Interval Test # f ‘(t) Sign of f ‘(t)

(-∞,-1) t = -2 24 +

(-1,2) t = 0 -12 -

(2,∞) t = 3 24 +

This tells us that in this interval the function is increasing

This tells us that in this interval the function is decreasing

This tells us that in this interval the function is increasing

Key Topics

• Homework: pg. 186 1 – 22 all