Let f(x, y) = xy + x − y be defined on the closed disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} of radius 2.(a) Find the maximum and minimum of Duf at (0, 0) over all unit vectors u.(b) Find the maximum and minimum of Duf over all points in the disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} and all unit vectors u. (Hint: Think of |∇f|2 as a function of x and y in the disk.)

Question
Asked Mar 12, 2019
44 views

Let f(x, y) = xy + x − y be defined on the closed disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} of radius 2.

(a) Find the maximum and minimum of Duf at (0, 0) over all unit vectors u.

(b) Find the maximum and minimum of Duf over all points in the disk {(x, y) ∈ R2 : x2 + y2 ≤ 4} and all unit vectors u. (Hint: Think of |∇f|2 as a function of x and y in the disk.)

check_circle

Expert Answer

Step 1

(a)

The function is defined on the closed disk of radius 2, that is

fullscreen
Step 2

Find the gradient of f and the unit vectors as follows.

fullscreen
Step 3

The maximum value of the directional derivative of ...

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Calculus