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Absolute extrema on a closed region practice problems

Absolute maximum and minimum on a closed rectangle: interior critical points, edges and corners. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} - x y - x - y^{2} - 2 y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -2 \le y \le 3 \).

Problem 10.299medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} + 3 x + y^{2} - 2 y \) on the rectangle \( \displaystyle -1 \le x \le 1 \), \( \displaystyle -1 \le y \le 1 \).

Problem 10.300medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} + x y - 4 x - 2 y^{2} + 4 y \) on the rectangle \( \displaystyle -1 \le x \le 3 \), \( \displaystyle -2 \le y \le 2 \).

Problem 10.301medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 2 x + 2 y^{2} \) on the rectangle \( \displaystyle -2 \le x \le 3 \), \( \displaystyle -2 \le y \le 1 \).

Problem 10.302medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 3 x - y^{2} + 3 y \) on the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle 0 \le y \le 3 \).

Problem 10.303medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - x^{2} + 2 x - 2 y^{2} \) on the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle 0 \le y \le 1 \).

Problem 10.304medium✓ Nihil obstat

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = x^{2} - x y + 2 x + 2 y^{2} - 4 y \) on the rectangle \( \displaystyle 0 \le x \le 1 \), \( \displaystyle -1 \le y \le 2 \).

Problem 10.305medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + 3 x + y^{2} - 2 y \) on the rectangle \( \displaystyle 0 \le x \le 2 \), \( \displaystyle 0 \le y \le 1 \).

Problem 10.306medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + x - y^{2} - 4 y \) on the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle -1 \le y \le 1 \).

Problem 10.307medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x, y) = - 2 x^{2} + x y - y^{2} + y \) on the rectangle \( \displaystyle 0 \le x \le 3 \), \( \displaystyle 0 \le y \le 1 \).

Problem 10.308medium✓ Every equation proved