APPLICATIONS OF DERIVATIVES CHAPTER: 4 APPLICATIONS OF DERIVATIVES

Sec 4.1: Extreme Values of Functions Outline: What are global and local maximum, minimum? Why? --------The extreme value theorem How? -------- How to find max and min value? Fermat’s theorem Critical points The closed interval method 2

Sec 4.1: Extreme Values of Functions absolute maximum global maximum local maximum relative maximum How many local maximum ??

Sec 4.1: Extreme Values of Functions local minimum relative minimum absolute minimum global minimum How many local minimum ??

Sec 4.1: Extreme Values of Functions The number f(c) is called the maximum value of f on D f(c) c d f(d) The number f(d) is called the minimum value of f on D The maximum and minimum values of f are called the extreme values of f.

Sec 4.1: Extreme Values of Functions EXAMPLE: EXAMPLE:

Sec 4.1: Extreme Values of Functions EXAMPLE: EXAMPLE:

Sec 4.1: Extreme Values of Functions local max at a,b,c REMARK: A function f has a local maximum at the endpoint b if for all x in some half-open interval

Sec 4.1: Extreme Values of Functions local min at a,b,c REMARK: A function f has a local minimum at the endpoint a if for all x in some half-open interval

Sec 4.1: Extreme Values of Functions

1 2 Sec 4.1: Extreme Values of Functions We have seen that some functions have extreme values, whereas others do not. 1 f(x) is continuous on 2 Closed interval [a, b] attains an absolute maximum f(c) and minimum f(d) value THE EXTREME VALUE THEOREM

1 2 Sec 4.1: Extreme Values of Functions THE EXTREME VALUE THEOREM f(x) is continuous on 2 Closed interval [a, b] attains an absolute maximum f(c) and minimum f(d) value THE EXTREME VALUE THEOREM Max?? Min?? What cond?? Max?? Min?? What cond??

Sec 4.1: Extreme Values of Functions f(x) is continuous on 2 Closed interval [a, b] attains an absolute maximum f(c) and minimum f(d) value THE EXTREME VALUE THEOREM Remark: The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. 13

Sec 4.1: Extreme Values of Functions interior point end point end point

Sec 4.1: Extreme Values of Functions

Sec 4.1: Extreme Values of Functions Remark: If f has a local maximum or minimum value at an interior point c then c is critical.

Sec 4.1: Extreme Values of Functions The only places where a function ƒ can possibly have an extreme value (local or global) are interior points where ƒ’= 0 interior points where is ƒ’ undefined, 3. endpoints of the domain of ƒ.

Sec 4.1: Extreme Values of Functions How to find absolute Max and Min Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) Evaluate ƒ at all criticals Evaluate ƒ at endpoints a and b Take the largest value and the smallest

Sec 4.1: Extreme Values of Functions How to find absolute Max and Min Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) Evaluate ƒ at all criticals Evaluate ƒ at endpoints a and b Take the largest value and the smallest

Sec 4.1: Extreme Values of Functions How to find absolute Max and Min Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) Evaluate ƒ at all criticals Evaluate ƒ at endpoints a and b Take the largest value and the smallest

Sec 4.1: Extreme Values of Functions EXAMPLE: Find: Local max and min Global max and min How to find absolute Max and Min Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) Evaluate ƒ at all criticals Evaluate ƒ at endpoints a and b Take the largest value and the smallest

Sec 4.1: Extreme Values of Functions EXAMPLE: EXAMPLE: Find: Absolute max and min Find: Absolute max and min

Sec 4.1: Extreme Values of Functions How to find absolute Max and Min (not closed interval) How to find absolute Max and Min Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) Evaluate ƒ at all criticals Evaluate ƒ at endpoints a and b Take the largest value and the smallest EXAMPLE: Find: Absolute max and min

Sec 4.1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min EXAMPLE: Find: Absolute max and min Remarks: Polynomial with odd degree  no absolute max, no absoulte min Polynomial with even degree  absolute max only or absoulte min only

How many local maximum How many local minimum Sec 4.1: Extreme Values of Functions How many local maximum How many local minimum

Sec 4.1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min EXAMPLE: Find: Absolute max and min Remarks: Polynomial with odd degree  no absolute max, no absoulte min Polynomial with even degree  absolute max only or absoulte min only

Sec 4.1: Extreme Values of Functions

Sec 4.1: Extreme Values of Functions

Sec 4.1: Extreme Values of Functions

Sec 4.1: Extreme Values of Functions