2. Consider the function f(x, y) = x³ + 3x² – 3xy? + 6xy – 3y² + 6y + 98. (a) Find all critical points (x, Y, f(x, y)) of the function f. - (b) Find A = fræfyy – f2, for any points found in part (a) and hence provide a conclusion (if possible) on their nature. - (c) This section considers an alternative method of investigating the nature of any critical points that remain unclassified and hence diagnose their nature. Consider what you do when the second derivative provides no classification in- formation in the case of functions of one variable and then think about how you might modify your investigation when the function now has two variables. For any critical points that remain unclassified, construct two tables: one where we choose an x-value to the left and right of the critical value, and one where we choose a y-value to the left and right of the critical value. For this assignment pick values that are 1 integer to the left and right. This yields a function of one variable at each of these test values that can be analysed for its shape. Thinking about how the shapes join together, a diagnosis on whether the critical point in question is a local maximum, minimum or saddle should emerge.
2. Consider the function f(x, y) = x³ + 3x² – 3xy? + 6xy – 3y² + 6y + 98. (a) Find all critical points (x, Y, f(x, y)) of the function f. - (b) Find A = fræfyy – f2, for any points found in part (a) and hence provide a conclusion (if possible) on their nature. - (c) This section considers an alternative method of investigating the nature of any critical points that remain unclassified and hence diagnose their nature. Consider what you do when the second derivative provides no classification in- formation in the case of functions of one variable and then think about how you might modify your investigation when the function now has two variables. For any critical points that remain unclassified, construct two tables: one where we choose an x-value to the left and right of the critical value, and one where we choose a y-value to the left and right of the critical value. For this assignment pick values that are 1 integer to the left and right. This yields a function of one variable at each of these test values that can be analysed for its shape. Thinking about how the shapes join together, a diagnosis on whether the critical point in question is a local maximum, minimum or saddle should emerge.
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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Transcribed Image Text:2. Consider the function f(x, y) = x³ + 3x² – 3xy² + 6xy – 3y² + 6y + 98.
(a) Find all critical points (x, Y, f(x, y)) of the function f.
(b) Find A
conclusion (if possible) on their nature.
fæz fyy - fa, for any points found in part (a) and hence provide a
(c) This section considers an alternative method of investigating the nature of any
critical points that remain unclassified and hence diagnose their nature.
Consider what you do when the second derivative provides no classification in-
formation in the case of functions of one variable and then think about how you
might modify your investigation when the function now has two variables. For any
critical points that remain unclassified, construct two tables: one where we choose
an x-value to the left and right of the critical value, and one where we choose a
y-value to the left and right of the critical value. For this assignment pick values
that are 1 integer to the left and right. This yields a function of one variable at
each of these test values that can be analysed for its shape. Thinking about how
the shapes join together, a diagnosis on whether the critical point in question is a
local maximum, minimum or saddle should emerge.
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