Question : 251) f(x, y) = xy, x2 + y2 = 72 A) : 1757206

251) f(x, y) = xy, x² + y² = 72

A) Maximum: 36 at (6, 6); minimum: -36 at (-6, -6)

B) Maximum: 36 at (6, -6) and (-6, 6); minimum: -36 at (6, 6) and (-6, -6)

C) Maximum: 36 at (6, 6) and (-6, -6); minimum: -36 at (6, -6) and (-6, 6)

D) Maximum: 36 at (6, 6); minimum: 0 at (0, 0)

252) f(x, y) = x² + y², xy² = 128

A) Maximum: none; minimum: 0 at (0, 0)

B) Maximum: 48 at (4, ±4 2); minimum: 0 at (0, 0)

C) Maximum: none; minimum: 48 at (4, ±4 2)

D) Maximum: 48 at (4, 4 2); minimum: -48 at (4, -4 2)

253) f(x, y) = y² - x², x² + y² = 9

A) Maximum: 9 at (0, ±3); minimum: -9 at (±3, 0)

B) Maximum: 18 at (0, ±3 2); minimum: -18 at (±3 2, 0)

C) Maximum: 18 at (0, ±3 2); minimum: -9 at (±3, 0)

D) Maximum: 9 at (0, ±3); minimum: -18 at (±3 2, 0)

254) f(x, y) = 4x + 6y, x² + y² = 13

A) Maximum: 36 at (3, 4); minimum: 0 at (0, 0)

B) Maximum: 26 at (2, 3); minimum: -26 at (-2, -3)

C) Maximum: 36 at (3, 4); minimum: -36 at (-3, -4)

D) Maximum: 26 at (2, 3); minimum: 0 at (0, 0)

255) f(x, y) = x²y, x² + 2y² = 6

A) Maximum: 18 at (3, 2); minimum: -18 at (-3, -2)

B) Maximum: 4 at (2, 1); minimum: -4 at (-2, -1)

C) Maximum: 4 at (±2, 1); minimum: -4 at (±2, -1)

D) Maximum: 18 at (±3, 2); minimum: -18 at (±3, -2)

256) f(x, y) = 3x - y + 1, 3x² + y² = 9

257) f(x, y) = x² + 4y³, x² + 2y² = 2

A) Maximum: 8 at (2, 1); minimum: -4 at (0, -1)

B) Maximum: 8 at (2, 1); minimum: -31 at (1, -2)

C) Maximum: 4 at (0, 1); minimum: -31 at (1, -2)

D) Maximum: 4 at (0, 1); minimum: -4 at (0, -1)

258) f(x, y) = xy, 9x² + 4y² = 36

259) f(x, y) = 12x + 3y, xy = 4, x > 0, y > 0

A) Maximum: 30 at (2, 2); minimum: 24 at (1, 4)

B) Maximum: 51 at (4, 1); minimum: 30 at (2, 2)

C) Maximum: none; minimum: 24 at (1, 4)

D) Maximum: none; minimum: 30 at (2, 2)

260) f(x, y, z) = x³ + y³ + z³, x² + y² + z² = 4

A) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: 0 at (0, 0, 0)

B) Maximum: 8 at (2, 0, 0); minimum: 0 at (0, 0, 0)

C) Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2)

D) Maximum: 8 at (2, 0, 0); minimum: -8 at (-2, 0, 0)