d i E Relative Extrema a l l u d Dr. Abdulla Eid b A - PowerPoint PPT Presentation

Section 13.1 d i E Relative Extrema a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Extrema 1 / 22

Application of Differentiation d One of the most important applications of differential calculus are the i E optimization problems , i.e., finding the optimal (best) way to do a something.In our case, these optimization problem are reduced to find the l l minimum or maximum of a function. u d b Example A 1 Find the quantity that maximizes the revenue. . r 2 Find the quantity that at least gives a return. D Dr. Abdulla Eid (University of Bahrain) Extrema 2 / 22

1 - Monotone Functions Increasing Function d i Algebra E Geometry a l If a ≤ b , then f ( a ) ≤ f ( b ) l u d b A Exercise . Write a similar definition for decreasing function. r D Definition A monotone function is either an increasing or decreasing function. Dr. Abdulla Eid (University of Bahrain) Extrema 3 / 22

d Question: How to tell when a function is increasing or decreasing? i E Answer: One way is to use the definition above, which is hard to do in a general. The other way is to use Calculus as follows: l l u If f ′ ( x ) ≥ 0, then f ( x ) is increasing. d b If f ′ ( x ) ≤ 0, then f ( x ) is decreasing. A . r D Dr. Abdulla Eid (University of Bahrain) Extrema 4 / 22

2 - Absolute Extrema Absolute Maximum (Global Maximum) Algebra d f ( c ) is an absolute maximum Geometry i E ( global maximum ) if a l l f ( x ) ≤ f ( c ) , for all x u d b A f ( c ) is the absolute maximum (only one). . c is called absolute maximizer r D Exercise Write a similar definition for absolute minimum . Definition An absolute extrema is either an absolute maximum or absolute minimum function. Dr. Abdulla Eid (University of Bahrain) Extrema 5 / 22

3 - Relative Extrema Relative Maximum (Local Maximum) Algebra d Geometry f ( c ) is an local maximum ( relative i E maximum ) if a l l u f ( x ) ≤ f ( c ) , for some value of x near d b f ( c ) is the local maximum (maybe more than one). A c is called local maximizer . r D Exercise Write a similar definition for local minimum . Definition An local extrema ( relative extrema is either an local maximum or loca minimum function. Dr. Abdulla Eid (University of Bahrain) Extrema 6 / 22

Example Let f ( x ) = x 2 , then d i It has a global minimum at (0,0). E a It has no global maximum. l l u d Example b A Let f ( x ) = e x , then . r It has no global minimum. D It has no global maximum. Dr. Abdulla Eid (University of Bahrain) Extrema 7 / 22

Critical Points Question: How to find the extrema (local min, local max, absolute min, d absolute max)? i E Answer: The following are the candidates for the extrma. a l l Definition u d A number c is called a critical point of f ( x ) if either b A f ′ ( c ) = 0 or f ′ ( c ) does not exist . r D Note: These critical points are the candidates for local maximum or local minimum. Dr. Abdulla Eid (University of Bahrain) Extrema 8 / 22

Example Find the critical points of the following function f ( x ) = x 3 + x 2 − x Solution: d i We find the derivative first which is E f ′ ( x ) = 3 x 2 + 2 x − 1 a l l u To find the critical points, we find where the derivative equal to zero or d b does not exist. A . f ′ ( x ) does not exist f ′ ( x ) = 0 r D denominator = 0 numerator = 0 3 x 2 + 2 x − 1 = 0 1 = 0 Always False x = − 1 or x = 1 No Solution 3 Dr. Abdulla Eid (University of Bahrain) Extrema 9 / 22

Example (Old Final Exam Question) Find the critical points of the following function f ( x ) = 2 x 3 − 6 x + 11 d Solution: i E We find the derivative first which is a f ′ ( x ) = 6 x 2 − 6 l l u d To find the critical points, we find where the derivative equal to zero or b does not exist. A . f ′ ( x ) does not exist r f ′ ( x ) = 0 D denominator = 0 numerator = 0 1 = 0 6 x 2 − 6 = 0 Always False x = 1 or x = − 1 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 10 / 22

Example Find the critical points of the following function � 1 − x 2 f ( x ) = d Solution: i E We find the derivative first which is a − 2 x l f ′ ( x ) = √ l u 1 − x 2 2 d b To find the critical points, we find where the derivative equal to zero or A does not exist. . r D f ′ ( x ) = 0 f ′ ( x ) does not exist numerator = 0 denominator = 0 1 − x 2 = 0 − 2 x = 0 x = 0 x = 1 or x = − 1 Dr. Abdulla Eid (University of Bahrain) Extrema 11 / 22

Example Find the critical points of the following function x − 1 f ( x ) = x 2 − x + 1 d Solution: i E We find the derivative first which is a − x 2 + 2 x f ′ ( x ) = ( x 2 − x + 1 )( 1 ) − ( x − 1 )( 2 x − 1 ) l l u = ( x 2 − x + 1 ) 2 ( x 2 − x + 1 ) 2 d b To find the critical points, we find where the derivative equal to zero or A does not exist. . r D f ′ ( x ) = 0 f ′ ( x ) does not exist numerator = 0 denominator = 0 − x 2 + 2 x = 0 x 2 − x + 1 = 0 x = 0 or x = 2 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 12 / 22

First Derivative Test d Question: How to find the local min, local max? i E Theorem a l (First Derivative Test) l u d 1 If f ′ ( x ) changes from positive to negative as x increases, then f has a b local maximum at a. A 2 If f ′ ( x ) changes from negative to positive as x increases, then f has a . r D local minimum at a. Dr. Abdulla Eid (University of Bahrain) Extrema 13 / 22

Example Find the intervals where the function is increasing/decreasing and find all local max/min. f ( x ) = 2 x 3 + 3 x 2 − 36 x d Solution: i E We find the derivative first which is a f ′ ( x ) = 6 x 2 + 6 x − 36 l l u d To find the critical points, we find where the derivative equal to zero or b does not exist. A . f ′ ( x ) does not exist r f ′ ( x ) = 0 D denominator = 0 numerator = 0 1 = 0 6 x 2 + 6 x − 36 = 0 Always False x = 2 or x = − 3 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 14 / 22

Number Line d i E a l l u d b A . 1 f is increasing in ( − ∞ , − 3 ) ∪ ( 2, ∞ ) . r D 2 f is decreasing in ( − 3, 2 ) . 3 f has a local maximum at x = − 3 with value f ( − 3 ) = 66. 4 f has a local minimum at x = 2 with value f ( 2 ) = − 44. Dr. Abdulla Eid (University of Bahrain) Extrema 15 / 22

Example (Old Exam Question) Find the intervals where the function is increasing/decreasing and find all local max/min. f ( x ) = − x 4 + 4 x 3 + 5 d i E Solution: We find the derivative first which is a l l f ′ ( x ) = − 4 x 3 + 12 x 2 = − 4 x 2 ( x − 3 ) u d b To find the critical points, we find where the derivative equal to zero or A does not exist. . r D f ′ ( x ) does not exist f ′ ( x ) = 0 denominator = 0 numerator = 0 1 = 0 − 4 x 2 ( x − 3 ) = 0 Always False x = 0 or x = 3 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 16 / 22

Number Line d i E a l l u d b A . 1 f is increasing in ( − ∞ , 3 ) . r D 2 f is decreasing in ( 3, ∞ ) . 3 f has a local maximum at x = 3 with value f ( 3 ) = 32. 4 f has a no local minimum. Dr. Abdulla Eid (University of Bahrain) Extrema 17 / 22

Example (Old Exam Question) Find the intervals where the function is increasing/decreasing, find all local max/min, and sketch the graph of the function. f ( x ) = x 3 − 12 x + 3 d i E Solution: a We find the derivative first which is l l u f ′ ( x ) = 3 x 2 − 12 d b To find the critical points, we find where the derivative equal to zero or A does not exist. . r D f ′ ( x ) does not exist f ′ ( x ) = 0 denominator = 0 numerator = 0 1 = 0 3 x 2 − 12 = 0 Always False x = 2 or x = − 2 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 18 / 22

Number Line d i E a l l u d b A . 1 f is increasing in ( − ∞ , − 2 ) ∪ ( 2, ∞ ) . r D 2 f is decreasing in ( − 2, 2 ) . 3 f has a local maximum at x = − 2 with value f ( − 2 ) = 19. 4 f has a local minimum at x = 2 with value f ( 2 ) = − 13. Dr. Abdulla Eid (University of Bahrain) Extrema 19 / 22

Exercise (Old Exam Question) Find the intervals where the function is increasing/decreasing and find all local max/min. f ( x ) = − 2 x 3 + 6 x 2 − 3 d i Solution: E We find the derivative first which is a l f ′ ( x ) = − 6 x 2 + 12 x l u d b To find the critical points, we find where the derivative equal to zero or A does not exist. . r D f ′ ( x ) does not exist f ′ ( x ) = 0 denominator = 0 numerator = 0 1 = 0 − 6 x 2 + 12 x = 0 Always False x = 2 or x = 0 No Solution Dr. Abdulla Eid (University of Bahrain) Extrema 20 / 22

Number Line d i E a l l u d b A . 1 f is decreasing in ( − ∞ , 0 ) ∪ ( 2, ∞ ) . r D 2 f is increasing in ( 0, 2 ) . 3 f has a local maximum at x = 2 with value f ( − 2 ) = 5. 4 f has a local minimum at x = 0 with value f ( 0 ) = − 3. Dr. Abdulla Eid (University of Bahrain) Extrema 21 / 22

Exercise (Old Exam Question) Find the intervals where the function is increasing/decreasing and find all local max/min. d f ( x ) = x 3 − 6 x 2 + 9 x + 1 i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Extrema 22 / 22