Let ü = (u1, u2) be a unit vector in R2 and let f: R2R be defined by if (r, y) # (0,0), f(r, y) = {12 + y? if (r, 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.
Let ü = (u1, u2) be a unit vector in R2 and let f: R2R be defined by if (r, y) # (0,0), f(r, y) = {12 + y? if (r, 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.
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: R? → R be defined by
if (x, y) # (0, 0),
f(r, u) = {r2+ y²
if (r, y) = (0,0).
%3D
(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%2F8627b494-ca5d-49f1-9a9e-a33da262702d%2Ff957d1ad-c658-4912-826e-345820f9f2ae%2Fa0buvx8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let ū = (u1, u2) be a unit vector in R? and let f: R? → R be defined by
if (x, y) # (0, 0),
f(r, u) = {r2+ y²
if (r, y) = (0,0).
%3D
(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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