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).

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

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 3 +18 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

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

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

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

In summary … QUESTIONS: p.178-180 #7, 10, 12, 13