Explanation Given: f is continuous function and f ( a , b ) < 0 Proof: We have to prove that there exists a δ − neighborhood about ( a , b ) such that f ( x , y ) < 0 for every point ( x , y ) in neighborhood. Given that f ( x , y ) is continuous function then lim ⁡ ( x , y ) → ( a , b ) f ( x , y ) = f ( a , b ) < 0 which means that for each ε > 0 there corresponds δ > 0 such that, whenever 0 < ( x − a ) 2 + ( y − b ) 2 < δ . Let ε = | f ( a , b ) | 2 then f ( x , y ) < 0 for every point corresponding δ − neighborhood Because | f ( x , y ) − f ( a , b ) | < | f ( a , b ) | 2 ⇒ − | f ( a , b ) | 2</

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13.2, Problem 86E

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To Prove: If f is continuous function and f(a,b)<0 then there exists a δ− neighborhood about (a,b) such that f(x,y)<0 for every point (x,y) in neighborhood.

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Chapter 13.2, Problem 85E

Chapter 13.3, Problem 1E

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Calculus: Early Transcendental Functions

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To Prove: If f is continuous function and f ( a , b ) < 0 then there exists a δ − neighborhood about ( a , b ) such that f ( x , y ) < 0 for every point ( x , y ) in neighborhood.

Solution Summary: The author explains that if f is continuous function, then there exists a -delta - neighborhood about (a,b)right.

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To Prove: If f is continuous function and f(a,b)<0 then there exists a δ− neighborhood about (a,b) such that f(x,y)<0 for every point (x,y) in neighborhood.

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Chapter 13 Solutions

To Prove: If f is continuous function and f ( a , b ) &lt; 0 then there exists a δ − neighborhood about ( a , b ) such that f ( x , y ) &lt; 0 for every point ( x , y ) in neighborhood.

Solution Summary: The author explains that if f is continuous function, then there exists a -delta - neighborhood about (a,b)right.

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To Prove: If f is continuous function and f(a,b)<0 then there exists a δ− neighborhood about (a,b) such that f(x,y)<0 for every point (x,y) in neighborhood.

Want to see the full answer?

Chapter 13 Solutions

To Prove: If f is continuous function and f ( a , b ) < 0 then there exists a δ − neighborhood about ( a , b ) such that f ( x , y ) < 0 for every point ( x , y ) in neighborhood.