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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 ox and Sy of sufficiently small magnitude justify your conclusions in part a).

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 ox and Sy of sufficiently small magnitude justify your conclusions in part a).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Part b please
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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