MAT 1234 Calculus I Section 3.1 Maximum and Minimum Values
Next WebAssign 3.1 Quiz– 2.8
1 Minute… You can learn all the important concepts in 1 minute.
1 Minute… High/low points – most of them are at points with horizontal tangent
1 Minute… High/low points – most of them are at points with horizontal tangent. Highest/lowest points – at points with horizontal tangent or endpoints
1 Minute… You can learn all the important concepts in 1 minute. We are going to develop the theory carefully so that it works for all the functions that we are interested in. There are a few definitions…
Preview Definitions absolute max/min local max/min critical number Theorems Extreme Value Theorem Fermat’s Theorem The Closed Interval Method
Max/Min We are interested in max/min values Minimize the production cost Maximize the profit Maximize the power output
Definition (Absolute Max) c D
Definition (Absolute Min) c D
Definition
x Example 1 y Absolute max. Absolute min.
Definition (Local Max/Min)
x Example 1 y Local max. Local min.
Q&A An end point is not a local max/min, why?
The Extreme Value Theorem
Q&A Give 2 examples of functions on an interval that do not have absolute max value.
Example 2 (No abs. max/min)
The interval is not closed
How to find Absolute Max./Min.?
Fermat’s Theorem c x y
Q&A: T or F
Definition (Critical Number)
Critical Number (Translation) Critical numbers give all the potential local max/min values
Critical Number (Translation)
Example 3 Find the critical numbers of
Example 3 Find the critical numbers of
The Closed Interval Method
Example 4 Find the absolute max/min values of
Expectations: Formal Conclusion