Section 12.8 - Maximum/Minimum Problems 19-34. Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test todetermine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point.Confirm your results using a graphing utility. 23. f (z, у) —1* + 2y? – 4ry

Let c≡(a, b) be the critical point of the given function. Then we have ∂(x4+2y2-4xy)∂x(a, b)=0 and…

Answered: 19-34. Analyzing critical points Find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Section 12.8 - Maximum/Minimum Problems

19-34. Analyzing critical points Find the critical points of the following functions. Use the Second Derivative Test to
determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point.
Confirm your results using a graphing utility.

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Step 1

Let $c \equiv (a, b)$ be the critical point of the given function. Then we have ${\frac{\partial (x^{4} + 2 y^{2} - 4 x y)}{\partial x}}_{(a, b)} = 0 a n d {\frac{\partial (x^{4} + 2 y^{2} - 4 x y)}{\partial y}}_{(a, b)} = 0$ which further implies that ${(4 x^{3} - 4 y)}_{(a, b)} = 0 a n d {(4 y - 4 x)}_{(a, b)} = 0$ implying that $a^{3} - b = 0 and b - a = 0$ leading to $(a, b) \equiv (0, 0), (1, 1), (- 1, - 1)$ . Therefore the required critical points for the given function are (0, 0), (1, 1), and (-1, -1).

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