1. Critical Points and Extrema
3.1 Part I
Critical Points and Extrema

2. Objectives Determine the local or global extreme values of functions.
1) Find 𝑓′(𝑥).
2) Set 𝑓 ′ 𝑥 =0. Solve for x. Also, determine where 𝑓 ′ 𝑥 is undefined.
3) Plug the x-values you found in part (2) into 𝑓(𝑥).
4) If there is an interval, plug the x-values into 𝑓(𝑥).
5) Look at the graph to determine if your ordered pairs in parts (3) & (4) are maximums or minimums.

3. Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum value on the interval.

4. Local Extreme Value Theorem
If a function f has a local maximum or minimum value, then either
c is an endpoint

5. Critical Points
A point at the interior of a function f at which f ′ = 0 or f ′ does not exist is a critical point of f.

6. Example 1
Determine where the function f(x) graphed below appears to have a derivative that is undefined or zero. Next, classify each as a relative max or relative min.
𝑓 ′ 𝑥 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑, 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑎𝑥
𝑓′ 𝑥 =0, 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑖𝑛

7. Example 2
Find the critical points for the function f(x), then determine whether they are relative maximum or relative minimum values
𝑓 𝑥 = 𝑥 3 + 𝑥 2 −8𝑥+5
𝑓 ′ 𝑥 =3 𝑥 2 +2𝑥−8
𝑓 ′ 𝑥 = 𝑥+2 3𝑥−4
0= 𝑥+2 3𝑥−4
𝑥=−2, 𝑥= 4 3
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑀𝑎𝑥 𝑎𝑡 𝑥=−2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑀𝑖𝑛 𝑎𝑡 𝑥= 4 3

8. Example 3
Find the critical points for the function f(x), then determine whether they are relative maximum or relative minimum values (you can use your calculator for this).
𝑓 𝑥 =𝑥−4 𝑥
𝑓 ′ 𝑥 =1− 2 𝑥
𝑓 ′ 𝑥 =0→𝑥=4
𝑓′(𝑥) is undefined→𝑥=0
0,0 𝑖𝑠 𝑎 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚
4,−4 𝑖𝑠 𝑎 𝑚𝑖𝑛𝑖𝑚𝑢𝑚