Multiple choice

Given a function: (x, y) = 4x2 + 6y2 − 8x − 4y + 8 The optimal value of f(x, y)

  1. is a minimum equal to 10/3

  2. is a maximum equal to 10/3

  3. is a minimum equal to 8/3

  4. is a maximum equal to 8/3
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A Correct answer
Explanation

Find partial derivatives: df/dx = 8x - 8 = 0 => x = 1. df/dy = 12y - 4 = 0 => y = 1/3. The second derivative test (f_xx = 8, f_yy = 12, f_xy = 0) shows a minimum since f_xx > 0 and D = f_xx*f_yy - f_xy^2 = 96 > 0. Value at (1, 1/3) is 4(1)^2 + 6(1/3)^2 - 8(1) - 4(1/3) + 8 = 4 + 2/3 - 8 - 4/3 + 8 = 4 - 2/3 = 10/3.

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