1. 4.2 Critical Points, Local Maxima and Local Minima
For a function 𝑓 𝑥 , a critical number is a number, 𝑐, in the domain of 𝑓(𝑥) such that 𝑓 ′ (𝑥)=0 or is undefined. As a result (𝑐, 𝑓 𝑐 ) is called a critical point and usually corresponds to local or absolute extrema (max/mins).

2. Steps for finding Local Maximum and Minimum Values
Find the critical numbers, c, of the function.
All 𝑥 such that 𝑓 ′ 𝑥 =0.
All 𝑥 such that 𝑓 ′ 𝑥 is undefined.
Set up a chart with intervals and see whether the derivative (slope) is increasing or decreasing on either side of these numbers.
Left of c
Right of c
Conclusion
𝑓 ′ 𝑥 <0
𝑓 ′ 𝑥 >0
Local minimum
Local maximum
Neither

3. Find the local extrema of each of the following functions.
Example #1:
Find the local extrema of each of the following functions.
(a) 𝑦= 𝑥 4 −8 𝑥 𝑥 2
Interval
𝒙<𝟎
𝟎<𝒙<𝟑
𝒙>𝟑
Sign of y’
Is the function Increasing/decreasing?
Shape of the curve
𝑦 ′ = 4𝑥 3 −24 𝑥 2 +36𝑥
0= 4𝑥 3 −24 𝑥 2 +36𝑥
0=4𝑥( 𝑥 2 −6𝑥+9)
0=4𝑥 (𝑥−3) 2
𝑥= 0, 3

4. Find the local extrema of each of the following functions.
Example #1:
Find the local extrema of each of the following functions.
(b) 𝑦= 𝑥 3
Interval
𝒙<𝟎
𝒙>𝟎
Sign of y’
Is the function Increasing/decreasing?
Shape of the curve
𝑦 ′ =3 𝑥 2
0=3 𝑥 2
𝑥=0

5. Find the local extrema of each of the following functions.
Example #1:
Find the local extrema of each of the following functions.
(c) 𝑓(𝑥)= (𝑥+2) 2 3
𝑓′(𝑥)= 2 3 (𝑥+2) − 1 3
Interval
𝒙<−𝟐
𝒙>−𝟐
Sign of f’(x)
Is the function Increasing/decreasing?
Shape of the curve
𝑓′(𝑥)= 2 3 (𝑥+2) 1 3
𝑓′(𝑥)≠0 for ∀𝑥∈𝑅
𝑓 ′ −2 is undefined

6. Example #2:
Graphing a derivative given the graph of a polynomial function.
Consider the graph of 𝑦=𝑓 𝑥 , graph 𝑦=𝑓′ 𝑥 .

7. In summary …
QUESTIONS: p #7, 10, 12, 13