Question

Asked Mar 12, 2019

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

(a) Find the maximum and minimum of D_{u}f at (0, 0) over all unit vectors u.

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

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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