# 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.)

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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.)

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Step 1

(a)

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

Step 2

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

Step 3

The maximum value of the directional derivative of ...

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