Critical Points and Extrema 3.1 Part I Critical Points and Extrema

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.

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.

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

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.

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

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

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