CHAPTER 4 APPLICATIONS OF DERIVATIVES Sec 4 1

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CHAPTER: 4 APPLICATIONS OF DERIVATIVES

CHAPTER: 4 APPLICATIONS OF DERIVATIVES

Sec 4. 1: Extreme Values of Functions Outline: • What are global and local

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

Sec 4. 1: Extreme Values of Functions absolute maximum global maximum local maximum relative

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

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

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:

Sec 4. 1: Extreme Values of Functions EXAMPLE:

Sec 4. 1: Extreme Values of Functions EXAMPLE:

Sec 4. 1: Extreme Values of Functions EXAMPLE:

Sec 4. 1: Extreme Values of Functions local max at a, b, c REMARK:

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:

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

Sec 4. 1: Extreme Values of Functions

Sec 4. 1: Extreme Values of Functions We have seen that some functions have

Sec 4. 1: Extreme Values of Functions We have seen that some functions have extreme values, whereas others do not. THE EXTREME VALUE THEOREM 1 2 f(x) is continuous on Closed interval [a, b] attains an absolute maximum f(c) and minimum f(d) value

Sec 4. 1: Extreme Values of Functions THE EXTREME VALUE THEOREM 1 2 f(x)

Sec 4. 1: Extreme Values of Functions THE EXTREME VALUE THEOREM 1 2 f(x) is continuous on Closed interval [a, b] Max? ? Min? ? What cond? ? attains an absolute maximum f(c) and minimum f(d) value Max? ? Min? ? What cond? ?

Sec 4. 1: Extreme Values of Functions THE EXTREME VALUE THEOREM 1 2 f(x)

Sec 4. 1: Extreme Values of Functions THE EXTREME VALUE THEOREM 1 2 f(x) is continuous on Closed interval [a, b] attains an absolute maximum f(c) and minimum f(d) value 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

Sec 4. 1: Extreme Values of Functions interior point end point

Sec 4. 1: Extreme Values of Functions

Sec 4. 1: Extreme Values of Functions

Sec 4. 1: Extreme Values of Functions Remark: If f has a local maximum

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

Sec 4. 1: Extreme Values of Functions The only places where a function ƒ can possibly have an extreme value (local or global) are 1. interior points where ƒ’= 0 2. 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

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals 3. Evaluate ƒ at endpoints a and b 4. Take the largest value and the smallest

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals 3. Evaluate ƒ at endpoints a and b 4. Take the largest value and the smallest

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals 3. Evaluate ƒ at endpoints a and b 4. Take the largest value and the smallest

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: 1) Local max and min

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: 1) Local max and min 2) Global max and min How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals 3. Evaluate ƒ at endpoints a and b 4. Take the largest value and the smallest

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min

Sec 4. 1: Extreme Values of Functions How to find absolute Max and Min 1. Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) 2. Evaluate ƒ at all criticals 3. Evaluate ƒ at endpoints a and b 4. Take the largest value and the smallest EXAMPLE: Find: Absolute max and min (not closed interval) How to find absolute Max and Min

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min Remarks:

Sec 4. 1: Extreme Values of Functions 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 1)How many local maximum 2)How many local

Sec 4. 1: Extreme Values of Functions 1)How many local maximum 2)How many local minimum

Sec 4. 1: Extreme Values of Functions EXAMPLE: Find: Absolute max and min Remarks:

Sec 4. 1: Extreme Values of Functions 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

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

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