a) Find all critical points of the function f(x,y) = x³ + 3y - y³ - 3x and classify them as local minima, local maxima or saddle points.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.7: More On Inequalities
Problem 44E
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a) Find all critical points of the function
f(x,y) = x³ + 3y - y³ - 3x
and classify them as local minima, local maxima or saddle points.
b) For each critical point (xo, Yo) you have identified in part a) above, calculate the Taylor
series expansion of f(xo + 8x, yo + dy) about the point (xo, Yo) up to (and including)
second-order terms.
By considering in each case the sign of f(xo + 8x, yo + dy)-f(xo, Yo) for all 8x and
Sy of sufficiently small magnitude justify your conclusions in part a).
Page 4 of 6
Transcribed Image Text:a) Find all critical points of the function f(x,y) = x³ + 3y - y³ - 3x and classify them as local minima, local maxima or saddle points. b) For each critical point (xo, Yo) you have identified in part a) above, calculate the Taylor series expansion of f(xo + 8x, yo + dy) about the point (xo, Yo) up to (and including) second-order terms. By considering in each case the sign of f(xo + 8x, yo + dy)-f(xo, Yo) for all 8x and Sy of sufficiently small magnitude justify your conclusions in part a). Page 4 of 6
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