Question : 231) f(x, y) = x2 + 8xy + y2 A) f(8, : 1757204

231) f(x, y) = x² + 8xy + y²

A) f(8, 8) = 640, local maximim

B) f(0, 0) = 0, saddle point

C) f(8, 0) = 64, local minimum; f(0, 8) = 64, local minimum

D) f(0, 0) = 0, saddle point; f(8, 8) = 640, local maximim

232) f(x, y) = x³ + y³ - 108x - 27y - 3

A) f(6, -3) = -381, saddle point; f(-6, 3) = 375, saddle point

B) f(-6, -3) = 483, local maximum; f(6, 3) = -489, local minimum

C) f(-6, -3) = 483, local maximum

D) f(6, 3) = -489. local minimum; f(6, -3) = -381, saddle point; f(-6, 3) = 375, saddle point; f(-6, -3) = 483, local maximum

233) f(x, y) = 2 - x⁴y⁴

A) f(2, 0) = 2, saddle point; f(0, 2) = 2, saddle point

B) f(0, 0) = 2, local maximum; f(2, 2) = -254, local minimum

C) f(0, 0) = 2, local maximum

D) f(2, 2) = -254, local minimum

234) f(x, y) = 4x²y + 6xy²

235) f(x, y) = 25x² + 40xy + 64y²

A) f(0, 0) = 0, local minimum

B) f(5, 8) = 6321, saddle point; f(8, 5) = 4800, saddle point

C) f(40, 40) = 179,200, local maximum; f(0, 0) = 0, local minimum

D) f(40, 40) = 179,200, local maximum

236) f(x, y) = (x² - 16)2 + (y² - 64)2

A) f(0, 0) = 4352, local maximum; f(-4, -8) = 0, local minimum

B) f(0, 0) = 4352, local maximum; f(0, 8) = 256, saddle point; f(0, -8) = 256, saddle point; f(4, 0) = 4352, saddle point; f(4, 8) = 0, local minimum; f(4, -8) = 0, local minimum;

f(-4, 0) = 4096, saddle point; f(-4, 8) = 0, local minimum; f(-4, -8) = 0, local minimum

C) f(0, 0) = 4352, local maximum; f(0, 8) = 256, saddle point; f(4, 0) = 4096, saddle point;

f(4, 8) = 0, local minimum; f(-4, -8) = 0, local minimum

D) f(0, 0) = 4352, local maximum; f(4, 8) = 0, local minimum; f(4, -8) = 0, local minimum;

f(-4, 8) = 0, local minimum; f(-4, -8) = 0, local minimum

237) f(x, y) = (x² - 49)2 - (y² - 100)2

A) f(0, 10) = 2401, local maximum; f(0, -10) = 2401, local maximum; f(7, 0) = -10,000, local minimum; f(-7, 0) = -10,000, local minimum

B) f(0, 0) = -7599, saddle point; f(7, 10) = 0, saddle point; f(7, -10) = 0, saddle point; f(-7, 10) = 0, saddle point; f(-7, -10) = 0, saddle point

C) f(0, 0) = -7599, saddle point; f(0, 10) = 2401, local maximum; f(7, 0) = -10,000, local minimum

D) f(0, 0) = -7599, saddle point; f(0, 10) = 2401, local maximum; f(0, -10) = 2401, local maximum; f(7, 0) = -10,000, local minimum; f(7, 10) = 0, saddle point; f(7, -10) = 0, saddle point; f(-7, 0) = -10,000, local minimum; f(-7, 10) = 0, saddle point; f(-7, -10) = 0, saddle point

238) f(x, y) = 5x + 7y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1)

A) Absolute maximum: 7 at (0, 1); absolute minimum: 0 at (0, 0)

B) Absolute maximum: 5 at (1, 0); absolute minimum: 0 at (0, 0)

C) Absolute maximum: 12 at (1, 1); absolute minimum: 5 at (1, 0)

D) Absolute maximum: 7 at (0, 1); absolute minimum: 5 at (1, 0)

239) f(x, y) = 9x² + 10y² on the closed triangular region bounded by the lines y = x, y = 2x, and x + y = 6

A) Absolute maximum: 196 at (2, 4); absolute minimum: 0 at (0, 0)

B) Absolute maximum: 196 at (2, 4); absolute minimum: 171 at (3, 3)

C) Absolute maximum: 171 at (3, 3); absolute minimum: 76 at (2, 2)

D) Absolute maximum: 171 at (3, 3); absolute minimum: 0 at (0, 0)

240) f(x, y) = 6x² + 10y² on the disk bounded by the circle x² + y² = 9

A) Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 54 at (3, 0) and (-3, 0)

B) Absolute maximum: 54 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0)

C) Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0)

D) Absolute maximum: 144 at (3, 3); absolute minimum: 0 at (0, 0)