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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A: This is a problem related to inner product space. Please see the next step for details.

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