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 + y? if (x, y) = (0, 0). (a) Find Daf(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 R? and let f: R² → R be defined by if (x, y) # (0, 0), f (x, y) = x² + y? if (x, y) = (0, 0). (a) Find Daf(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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Question
![Let ū = (u1, u2) be a unit vector in R? and let f: R² → R be defined by
x²y
x2 + y?
if (x, y) # (0, 0),
f (x, y) =
if (x, y) = (0,0).
(a) Find Daf(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?]
(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 ||Vf(0,0)|| and –||Vf(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%2Fe9f8ab2f-3a8f-43b3-98b3-87543a4fbec0%2F4efefe4b-f699-4491-aa42-6c4433ee6a73%2F7gnex6_processed.png&w=3840&q=75)
Transcribed Image Text:Let ū = (u1, u2) be a unit vector in R? and let f: R² → R be defined by
x²y
x2 + y?
if (x, y) # (0, 0),
f (x, y) =
if (x, y) = (0,0).
(a) Find Daf(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?]
(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 ||Vf(0,0)|| and –||Vf(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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