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3. Consider the function f(r, y) = 4x² + 4y² – 4x + 4 defined on R². (i) Compute all critical points of f and classify them as either local maxima, local minima or saddle points. (ii) Compute the global maxima and global minima of f, if they exist. If they do not exist, explain why (i.e., prove it). (iii) Let D = {(r, y) € R² | r² + 4y² < 4} and assume f(r, y) is defined on D. Com- pute the global maxima and global minima and discuss their existence (use both approaches, i.e., the Lagrange Multipliers Principle and the analysis of f(x, y) on the boundary of D, ƏD).

3. Consider the function f(r, y) = 4x² + 4y² – 4x + 4 defined on R². (i) Compute all critical points of f and classify them as either local maxima, local minima or saddle points. (ii) Compute the global maxima and global minima of f, if they exist. If they do not exist, explain why (i.e., prove it). (iii) Let D = {(r, y) € R² | r² + 4y² < 4} and assume f(r, y) is defined on D. Com- pute the global maxima and global minima and discuss their existence (use both approaches, i.e., the Lagrange Multipliers Principle and the analysis of f(x, y) on the boundary of D, ƏD).

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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3. Consider the function f(x, y) = 4.x? + 4y? – 4x + 4 defined on R². (i) Compute all critical points of ƒ and classify them as either local maxima, local minima or saddle points. (ii) Compute the global maxima and global minima of f, if they exist. If they do not exist, explain why (i.e., prove it). (iii) Let D = {(x, y) € R² |x² + 4y? < 4} and assume f(x, y) is defined on D. Com- pute the global maxima and global minima and discuss their existence (use both approaches, i.e., the Lagrange Multipliers Principle and the analysis of f(x, y) on the boundary of D, ðD).
Transcribed Image Text:3. Consider the function f(x, y) = 4.x? + 4y? – 4x + 4 defined on R². (i) Compute all critical points of ƒ and classify them as either local maxima, local minima or saddle points. (ii) Compute the global maxima and global minima of f, if they exist. If they do not exist, explain why (i.e., prove it). (iii) Let D = {(x, y) € R² |x² + 4y? < 4} and assume f(x, y) is defined on D. Com- pute the global maxima and global minima and discuss their existence (use both approaches, i.e., the Lagrange Multipliers Principle and the analysis of f(x, y) on the boundary of D, ðD).
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