Question 2. Consider the function f(x, y) = x³-12xy + 8y³. (a) Find all the stationary points of the function f(x, y) and determine, if possible, whether each is a local maximum, local minimum or saddle point. (b) (i) Find the gradient vector of f(x, y) at (x, y) = (1,1). (ii) Hence, or otherwise, find the directional derivative of f(x,y) at (x,y) = (1,2) in the direction u = (iii) In what direction is f(x, y) decreasing most rapidly at the point (x, y) = (1, 1)?
Question 2. Consider the function f(x, y) = x³-12xy + 8y³. (a) Find all the stationary points of the function f(x, y) and determine, if possible, whether each is a local maximum, local minimum or saddle point. (b) (i) Find the gradient vector of f(x, y) at (x, y) = (1,1). (ii) Hence, or otherwise, find the directional derivative of f(x,y) at (x,y) = (1,2) in the direction u = (iii) In what direction is f(x, y) decreasing most rapidly at the point (x, y) = (1, 1)?
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:Question 2.
Consider the function f(x, y) = x³-12xy + 8y³.
(a)
Find all the stationary points of the function f(x, y) and determine, if possible,
whether each is a local maximum, local minimum or saddle point.
(b)
(i) Find the gradient vector of f(x, y) at (x, y) = (1,1).
(ii) Hence, or otherwise, find the directional derivative of f(x, y) at (x,y) = (1,2) in the
direction u =
(iii) In what direction is f(x, y) decreasing most rapidly at the point (x, y) = (1, 1)?
Expert Solution
Step 1
The saddle point is (0, 0)
Local minima is (2, 1)
Step by step
Solved in 2 steps with 1 images
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