誰可以幫我解這些微積分阿

Question

1.Find the local maxima, local minima, and saddle point for
f(x,y)=x四次方- 8x二次方+ 3y二次方- 6y

2.Find the local maxima, local minima, and saddle point for
f(x,y)=x三次方+ y三次方- 3xy + 5

Answer 1

1. f(x,y)=x4- 8x2+ 3y2- 6yfx(x,y)=4x3-16xfy(x,y)=6y-6f之critical points滿足4x3-16x=0和6y-6=0解得(x,y)=(0,1),(-2,1)和(2,1)又fxx(x,y)=12x2-16, fyy(x,y)=6, fxy(x,y)=0, fyx(x,y)=0fxx(0,1)fyy(0,1)-fxy(0,1)fyx(0,1)=(-16)(6)-(0)(0)=-96 < 0∴(0,1)為saddle pointfxx(-2,1)fyy(-2,1)-fxy(-2,1)fyx(-2,1)=(32)(6)-(0)(0)=192 > 0且fxx(-2,1)=32 > 0∴在(-2,1)有relative minimum f(-2,1)=(-2)4-8(-2)2+3(1)2-6(1)=-19fxx(2,1)fyy(2,1)-fxy(2,1)fyx(2,1)=(32)(6)-(0)(0)=192 > 0且fxx(2,1)=32 > 0∴在(2,1)有relative minimum f(2,1)=(2)4-8(2)2+3(1)2-6(1)=-192. f(x,y)=x3+ y3- 3xy + 5 fx(x,y)=3x2-3yfy(x,y)=3y2-3xf之critical points滿足3x2-3y=0和3y2-3x=0解得(x,y)=(0,0),(1,1)又fxx(x,y)=6x, fyy(x,y)=6y, fxy(x,y)=-3, fyx(x,y)=-3fxx(0,0)fyy(0,0)-fxy(0,0)fyx(0,0)=(0)(0)-(-3)(-3)=-9 < 0∴(0,0)為saddle pointfxx(1,1)fyy(1,1)-fxy(1,1)fyx(1,1)=(6)(6)-(-3)(-3)=36-9=27 > 0且fxx(1,1)=6 > 0∴在(1,1)有relative minimum f(1,1)= 13+13-3(1)(1)+5=4