sole11 - ANALYSIS EXERCISE 11SOLUTIONS 1. By the definition...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ANALYSIS EXERCISE 11SOLUTIONS 1. By the definition of continuity for every x [ a,b ] lim x x f ( x ) = f ( x ) . Using the Heine definition one can compute the above limit along the sequences x n x , x n 6 = x . Choose x n Q . Then f ( x n ) = 0 for all n N , and it is clear that f ( x ) = 0. 2. (a) Let f : R R be defined by f ( x ) = x 3- x + 3. Then f (- 2) =- 3 < 0 and f (0) = 3 > 0. By the intermediate value theorem ( x [- 2 , 0])( f ( x ) = 0). (b) Let f : R R be defined by f ( x ) = x 5 + x + 1. Then f (- 2) =- 33 < 0 and f (0) = 1 > 0. By the intermediate value theorem ( x [- 2 , 0])( f ( x ) = 0). 3. Let = f- g . Then is continuous. ( a ) = f ( a )- g ( a ) < 0, and ( b ) = f ( b )- g ( b ) > 0. Therefore by the intermediate value theorem ( x ( a,b ))( ( x ) = 0). 4. Let f : R R be defined by f ( x ) = x 4- 3 x- 9. f (- 2) = 13 > 0, f (2) = 1 > 0. But f (0) =- 9 < 0. So by the intermediate value theorem there is an x 1 (- 2 , 0) such that f ( x 1 ) = 0, and there is an x 2 (0 , 2) such that f ( x 2 ) = 0. 5. Let g : [0 , 1] R defined by g ( x ) = f ( x )- x . Then g (0) = f (0) 0. If g (0) = 0 then x = 0 is the desired solution. Now assume that g (0) > 0. g (1) = f (1)- 1 0. If g (1) = 0 then x = 1 is the desired solution. Next assume that g (1) < 0. Then by the intermediate value theorem ( x (0 , 1))(...
View Full Document

This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Winter '10 term at Bristol Community College.

Page1 / 3

sole11 - ANALYSIS EXERCISE 11SOLUTIONS 1. By the definition...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online