Question

please do 9,13,17 only

8. What is the procedure for locating absolute maximum and mi mum values on a closed bounded domain? Basic Skills 9-18. Critical points Find all critical points of the following functions. 9. f(x, y) = 1 + x² + y2 10. f(x, y) = x² - 6x + y² + 8y 11. f(x, y) = (3x – 2)2 + (y – 4)2 12. f(x, y) = 3x2 – 4y2 13. f(x, y) = x4 + y4 – 16xy 14. f(x, y) = x°/3 – y/3 + 3xy 15. f(x, y) = x4 – 2x² + y² – 4y + 5 16. f(x, y) = x² + xy - 2x – y + 1 17. f(x, y) = x2 + 6x + y2 + 8 18. f(x, y) = e*?y2– 2x y2 +.y?

Answer 1

f(x, y) = txty' . fr = = 22 fy = 24 fu= => 226=0 so x=o fya => 245o =) yzo so (o, os is the only critical point. 13 fra, y) = x 4+4-loxy for 4x² - 164 ty = 44016* 3 = 4y= uz reyeo 443 16x = 0 fy 4 (23216250 16x =o → 748 TE (6)=0 8 -16=0 16 = 256 2 - x-2 or for x = 0, y = 0 for x=2, y=2 y=2 n=-2, for (0,0) (2,2) & (-2,-2) are the critical pants.

of(x,y) = 5e²+ 6x + y²+8 fx = 2xto fy = 2 2y fx=0 = 2x+6=0 x=-3 fyzo - 2y =o soo (-3,0) is the critical point of flacy)