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Show that (0,0) is a critical point of f(x,y) = x kxy + y2 no matter what value the constant k has

Show that (0,0) is a critical point of f(x,y) = x kxy + y2 no matter what value the constant k has

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Show that (0,0) is a critical point of f(x,y) = x + kxy + y no matter what value the constant k has Choose the correct definition of a critical point below. O A. A critical point occurs when fyx = fyy # 0. O B. A critical point occurs either when fy = fy = 0, or when f, or fy, fails to exist. OC. A critical point occurs when fx = fy # 0. O D. A critical point occurs either when fyx = fyy = 0, or when fyx or fyy fails to exist. Determine fy. Determine fy. Evaluate fy at (0,0). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. (0,0) = (Type an expression using k as the variable. Simplify your answer.) O B. fx(0,0) does not exist.
Transcribed Image Text:Show that (0,0) is a critical point of f(x,y) = x + kxy + y no matter what value the constant k has Choose the correct definition of a critical point below. O A. A critical point occurs when fyx = fyy # 0. O B. A critical point occurs either when fy = fy = 0, or when f, or fy, fails to exist. OC. A critical point occurs when fx = fy # 0. O D. A critical point occurs either when fyx = fyy = 0, or when fyx or fyy fails to exist. Determine fy. Determine fy. Evaluate fy at (0,0). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. (0,0) = (Type an expression using k as the variable. Simplify your answer.) O B. fx(0,0) does not exist.
Evaluate fy at (0,0). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. fy(0,0) = | (Type an expression using k as the variable. Simplify your answer.) O B. y(0,0) does not exist. Is this information sufficient to show that (0,0) satisfies the definition of a critical point for f(x,y), for every value of the constant k? O A. No, because both of the partial derivatives are not equal to zero at (0,0) for some values of k. O B. No, because at least one of the partial derivatives is not equal to zero at (0,0) for all values of k. C. Yes, because both of the partial derivatives are undefined at (0,0) for all values of k. O D. Yes, because both partial derivatives are equal to zero at (0,0) for all values of k.
Transcribed Image Text:Evaluate fy at (0,0). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. fy(0,0) = | (Type an expression using k as the variable. Simplify your answer.) O B. y(0,0) does not exist. Is this information sufficient to show that (0,0) satisfies the definition of a critical point for f(x,y), for every value of the constant k? O A. No, because both of the partial derivatives are not equal to zero at (0,0) for some values of k. O B. No, because at least one of the partial derivatives is not equal to zero at (0,0) for all values of k. C. Yes, because both of the partial derivatives are undefined at (0,0) for all values of k. O D. Yes, because both partial derivatives are equal to zero at (0,0) for all values of k.
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