1. Increasing and Decreasing Functions
Recall again :)
If f ' > 0 on an interval, the f is increasing on the interval
If f ' < 0 on an interval, the f is decreasing on the interval
Ex. Given f(x) = x3 + 3x2 - 24x + 18. Determine the
intervals on which f(x) is increasing or decreasing.
Step 1: Find f '(x)
Step 2: Factor the derivative
Step 3: Find the critical numbers
Recall Critical point = any value in the domain where
f '(c) = 0 or f '(x) does not exist.
Step 4: Plot the critical numbers on a number line to
divide the function into intervals
Step 5: Test each interval to determine if f '(x) is positive
or negative and determine if f is increasing or
decreasing on the intervals

2. Recall, graphically critical points occur where:
- a tangent is horizontal f '(x) = 0
- the graph has a sharp corner or cusp
- a tangent is vertical f '(x) is undefined
cusp
corner

3. Ex. Find the critical numbers for the given functions:
ex
a)
x-2
b) x3/5(4 - x)

4. Given that c is a critical number of a function f:
f has a local maximum at x = c if for all x near c, f(c) ≥ f(x)
f has a local minimum at x = c if for all x near c, f(c) ≤ f(x)
a
c
d
b
global maximum = highest point over entire domain of f
global minmum = lowest point over entire domain of f
*can only have one of each!

5. First Derivative Test
If c is a critical number and if f ' changes sign at c then
- f has a local minimum at x = c if f ' is negative to the left
of c and positive to the right of c
- f has a local maximum at x = c if f ' is positive to the left
of c and negative to the right of c
Local Maximum
Local Minimum

6. Ex. Use the first derivative test to find the local maximum
and minimum values of f if f(x) = x4 + 2x3 + x2 - 8