Let ū = (u1, u2) be a unit vector in R2 and let f: R2 → R be defined by if (x, y) # (0,0), f(x, y) = { x2 + y² if (x, y) = (0,0). (a) Find Dif(0,0). (b) Using your solution to (a), find Vf(0,0). (c) Use the Lagrange multipliers algorithm to find the maximum and minimum directional derivatives at (0,0). [Hint: What are you trying to optimize? What is the constraint?]
Let ū = (u1, u2) be a unit vector in R2 and let f: R2 → R be defined by if (x, y) # (0,0), f(x, y) = { x2 + y² if (x, y) = (0,0). (a) Find Dif(0,0). (b) Using your solution to (a), find Vf(0,0). (c) Use the Lagrange multipliers algorithm to find the maximum and minimum directional derivatives at (0,0). [Hint: What are you trying to optimize? What is the constraint?]
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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Please take your time and solve correctly by hand solution
![Q4. Let ū = (u1, u2) be a unit vector in R? and let f: R2 → R be defined by
if (x, y) # (0,0),
f(x, y) =
x2 + y2
if (x, y) = (0,0).
(a) Find Daf(0,0).
(b) Using your solution to (a), find Vƒ(0,0).
(c) Use the Lagrange multipliers algorithm to find the maximum and minimum directional
derivatives at (0,0). [Hint: What are you trying to optimize? What is the constraint?]
(d) If you've solved (b) and (c) correctly, you will have found that the maximum and mini-
mum directional derivatives are not equal to ||V f(0, 0)|| and –||V f(0, 0)||. This appears
to contradict the Greatest Rate of Change Theorem given in Unit 7.2. What went wrong?
Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0719dc05-02f0-4a27-95ad-860ae24ff415%2F27f0a275-d6b7-49e0-a252-4c1865c941eb%2Fbyzz2oq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q4. Let ū = (u1, u2) be a unit vector in R? and let f: R2 → R be defined by
if (x, y) # (0,0),
f(x, y) =
x2 + y2
if (x, y) = (0,0).
(a) Find Daf(0,0).
(b) Using your solution to (a), find Vƒ(0,0).
(c) Use the Lagrange multipliers algorithm to find the maximum and minimum directional
derivatives at (0,0). [Hint: What are you trying to optimize? What is the constraint?]
(d) If you've solved (b) and (c) correctly, you will have found that the maximum and mini-
mum directional derivatives are not equal to ||V f(0, 0)|| and –||V f(0, 0)||. This appears
to contradict the Greatest Rate of Change Theorem given in Unit 7.2. What went wrong?
Explain.
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