(1) Consider the function f(x, y) = x³ - 2xy - y². (i) Verify that (0,0) and (-2/3, 2/3) are the critical points of f. (ii) Compute the Hessian H(f) = farfyy-fry of f at each critical point of f. (iii) Locate a saddle point of f and a local maximum point of f.

(1) Consider the function f(x, y) = x³ - 2xy - y². (i) Verify that (0,0) and (-2/3, 2/3) are the critical points of f. (ii) Compute the Hessian H(f) = farfyy-fry of f at each critical point of f. (iii) Locate a saddle point of f and a local maximum point of f.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter14: Discrete Dynamical Systems

Section14.3: Determining Stability

Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...

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Question

Calc

(1) Consider the function f(x, y) = x³ - 2xy - y².
(i) Verify that (0,0) and (-2/3, 2/3) are the critical points of f.
(ii) Compute the Hessian H(f) = farfyy - fy of f at each critical point of f.
(iii) Locate a saddle point of f and a local maximum point of f.
(iv) Use a cmputer to sketch the level curve f(x, y) = 0 and indicate the regions of R²
where f(x, y) is positive and where it is negative.

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Step 1: Define the problem

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Step 2: Calculate the first order partial derivatives of function f(x,y,z)

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Step 3: Calculate the critical points

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Step 4: Calculate the second order partial derivatives

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Step 5: Calculate the Hessian at critical points

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Step 6: Calculate the location of saddle point and local maximum point of f

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