Section 3.1 Day 1 Extrema on an Interval AP Calculus AB

Learning Targets Define the terms absolute extrema, relative extrema, absolute maximum/minimum, relative maximum/minimum, critical number Determine the absolute extrema and relative extrema from a graph Determine the absolute extrema analytically of a function Define and apply the Extreme Value Theorem Determine the critical numbers of a function Communicate the relationship between the relative extrema and critical numbers

The maximum function value on the given interval Extrema Absolute Maximum The maximum function value on the given interval Extrema The minimum & maximum values of the function on the given interval Absolute Minimum The minimum function value on the given interval

The maximum function value on the open interval Relative Extrema Relative Maximum The maximum function value on the open interval Relative Extrema The minimum & maximum values of the function on an open interval Relative Minimum The minimum function value on the open interval

Key Point It is possible for a function to not have a maximum or a minimum. For example… Absolute Extrema: the function could have a hole at those points Relative Extrema: the function could be continuously increasing or decreasing over the entire real number line

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

Relative Extrema In your groups, use the knowledge you have learned from calculus this year to determine what is the mathematical similarity between the relative extrema on this graph.

Critical Numbers Let 𝑓 be defined at 𝑐. If 𝑓 ′ 𝑐 =0 or if 𝑓 is not differentiable at 𝑐, then 𝑐 is a critical number of 𝑓.

Relative Extrema & Critical Numbers If 𝑓 has a relative extrema at 𝑥=𝑐, then 𝑐 is a critical number of 𝑓. In other words, relative extrema only occur at critical numbers.

Example 1 Determine if the graph will consist of both absolute and relative extrema or just relative extrema. Support your stance. Determine the coordinate points of the extrema of the function

Example 2 Determine if the graph will consist of both absolute and relative extrema or just relative extrema. Support your stance. Determine the coordinate points of the extrema of the function

Procedure for finding Absolute Extrema on a Closed Interval Find the critical numbers of 𝑓 in the interval Evaluate 𝑓 at each critical number in the interval Evaluate 𝑓 at the endpoints of the interval Determine where the absolute extrema occur at

Example 3 Find the absolute extrema of 𝑓 𝑥 =3 𝑥 2 −4 𝑥 3 on the interval [−1, 2] 1. 𝑓 ′ 𝑥 =6𝑥−12 𝑥 2 =6𝑥(1−2𝑥) 2. 𝑓 ′ 𝑥 =0 ⇒ 𝑥=0, 𝑥= 1 2 3. −1, 7 , 0, 0 , 1 2 , 1 4 , 2, −20 Min (2, −20) Max (−1, 7)

Example 4 Find the absolute extrema of 𝑓 𝑥 =2𝑥−3 𝑥 2 3 on the interval [−1, 3] 𝑓 ′ 𝑥 =𝑥−2 𝑥 − 1 3 𝑓 ′ 𝑥 =0⇒𝑥= 4 8 𝑓 ′ 𝑥 =𝐷𝑁𝐸⇒𝑥=0 Min (−1, −5) Max (0, 0)

Example 5 Find the absolute extrema of 𝑓 𝑥 = 𝑡 2 𝑡 2 +3 on the interval [−1, 1] Min (0, 0) Max −1, 1 4 , 1, 1 4

Exit Ticket for Feedback Find the absolute extrema of 𝑓 𝑥 =− 𝑥 2 +3𝑥 on the interval [0, 3]