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

See similar textbooks

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

Videos

Question

Chapter 13.2, Problem 86E

To determine

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.2, Problem 85E

Chapter 13.3, Problem 1E

Students have asked these similar questions

Mathmatic Let f : ℝ2 → ℝ be a continuous function, and let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + xy + 3y2 = 3. Show that f(S) contains a minimum and maximum value.

Prove , no typing solution

pdf.6 öyölas -> find (f-g)(x) (f+g)x) f.g)). ) Find (f/g)(x) and (8/f)(x) for the functions given by f(r) = r and g(x) = /4-. Then find the domains of f/lg and g/f. ★******* ******************************************************** Composition of Functions Definition of Composition of Two Functions- The composition of the function of f with the function g is: (fo g) (x) =f(g (x)). The domain of (fo g) is the set of all x in the domain of g such that g (x) is in the domain of f. For instance, iff (x)= x² and g (x) = x+1, the composition of f with g is: f(g (x)) = (x+1) Abe (Н.W) If f(x) = 4 - x² & g(x) = Vx then find (fog)(), (gof)(x) x+8 If f(x) = 3x - 8 & g(x) = then find (fog)(x). (gof)cx) 3 and is (fog)c).(gof) are equal ?? x2 - 2x & g(x) = 3-x then solve the equations: a) (fog)) = 0 & b) (gof)m + x +5: If f(x) = x - 9 & g(x) = 2x - 5 then find the solution for (fog)(x) <0 If fx)=x2-2x-3 & g(x) = 1X then find: a) (fog)(x) & b) (gof)) T F)%3 & g(x) = then find b) (gof)< x+2 a)…

Chapter 13 Solutions

Calculus: Early Transcendental Functions

Ch. 13.1 - Think About It Explain why z2=x+3y is not a...Ch. 13.1 - Function of Two Variables What is a graph of a...Ch. 13.1 - Determine whether graph is a function. Use the...Ch. 13.1 - Contour Map Explain how to sketch a contour map of...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Determining Whether an Equation Is a Function In...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...Ch. 13.1 - Evaluating a Function In Exercises 9-20, evaluate...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

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 ) &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.

Concept explainers

Videos

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.