14.6  :

1,
a)
fx=2xy^3,	fy=3x^2y^2,	fz=4z^3
b)
fxs=2s,	fys=t^2,	fzs=2st
c)
fs=4xy^3s+3x^2y^2t^2+8z^3st=7s^6t^6+8s^7t^4

3,
fx=y,	fy=x,	fz=2z,
fxs=2s,	fys=2r,	fzs=0
fxr=0,	fyr=2s,	fzr=2r
fs=2ys+x2r=6rs^2
fr=2sx+4zr=2s^3+4r^3

5,
fgx=-sin(x-y),	fgy=sin(x-y)
fxu=3,	fyu=-7
fxv=-5,	fyu=15
fgu=-10sin(10u-20v)
fgv=20sin(10u-20v)

7,
Fu=e^(u+v),	Fv=e^(u+v)
fuy=0,	fvy=x
Fy=xe^(x^2+xy)

15,
fgx=2x,	fgy=2y
fxu=e^u cos v,	fyu=e^u sin v
fgu=2(e^u cos v)^2+2(e^u sin v)^2
g(0,1)=2

23,
fxs=1,	fys=1
fxt=1,	fyt=-1
fs=fx+fy
ft=fx-fy
fs*ft=(fx+fy)(fx-fy)=fx^2-fy^2

27,
fzx=2xy+y^2(fzx)+z^2+x2z(fzx)=0
(y^2+2z)(fzx)=-2xy-z^2
fzx=(-2xy-z^2)/(y^2+x2z)

29,
xe^xy+x(fzy)cos(xz)+1=0
x(fzy)cos(xz)=-(xe^xy+1)
fzy=-(xe^xy+1)/(x cos(xz))

31
(-2w)(fwy)/(w^2+x^2)^2+(-2w*(fwy)-2y)/(w^2+y^2)^2=0
fwy(1,1,1)=-1/2

14.7  :

1,
a)
fx=2x-4y
2x-4y=0
x=2y
a=2b
fy=4y^3-4x
4y^3-4x=4y^3-8y=0
y=-sqrt2,0,sqrt2
P:(0,0),(2sqrt2,sqrt2),(-2sqrt2,-sqrt2)
b)
Local minima:(2sqrt2,sqrt2),(-2sqrt2,-sqrt2)
saddle point:(0,0)
absolute minimum value:f(-2sqrt2,-sqrt2)=-4

3,
fx=2x+y
fy=32y^3+x-6y-3y^2
2x+y=0,	32y^3+x-6y-3y^2=0
x=-1/4,0,13/64	y=1/2,0,-13/32
local minimum:(-1/4,1/2),(13/64,-13/32)
saddle point:(0,0)

5,
a)
fx=y^2-2xy+y,	fy=2yx-x^2+x
y^2-2xy+y=y(y-2x+1)=0,	2yx-x^2+x=x(2y-x+1)=0
b)
x=0,y=-1,		x=0,y=0,		x=1/3,y=-1/3,	x=1,y=0
c)
fxx=-2y
fxy=2y-2x+1
fyy=2x
d(0,-1)=-1<0
fxx(0,-1)>0
(0,-1)is saddle point
d(0,0)=-1
(0,0)is a saddle point
d(1/3,-1/3)=1/3>0
fxx(1/3,-1/3)>0
(1/3,-1/3)is local minimum
d(1,0)=-1<0
(1,0) is saddle point

7,
fx=2x-y+1
fy=2y-x
fxx=2
fxy=-1
fyy=2
2x-y+1=0,	2y-x=0
x=-2/3,	y=-1/3
d(-2/3,-1/3)=3>0
fxx(-2/3,-1/3)=2>0
(-2/3,-1/3)is local minimum

11,
fx=4-9x^2-2y^2
fy=-4xy
fxx=-18x
fxy=-4y
fyy=-4x
4-9x^2-2y^2=0,	-4xy=0
Critical Points:           (-2/3,0),	(0,-sqrt2),	(0,sqrt2),	(2/3,0)
d(-2/3,0)=32>0
fxx>0
(-2/3,0) is a local minimum.

13,
fx=4x^3-4y
fy=4y^3-4x
fxx=12x^2
fxy=-4
fyy=12y^2
4x^3-4y=0,	4y^3-4x=0
(-1,-1),	(0,0),	(1,1)
d(-1,-1)=128>0
fxx>0
(-1,-1)is a local minimum
d(0,0)=-16<0
(0,0)is a saddle point
d(1,1)=128>0
fxx>0
(1,1)is a local minimum

17,
fx=cos(x+y)+sin x
fy=cos(y+x)
fxx=-sin(x+y)+cos x
fxy=-sin(x+y)
fyy=cos(y+x)
cos(x+y)+sin x=0,	cos(y+x)=0
(0,-1.5708)
d(0,-1.5708)<0
(0,1.5708) is a saddle point.

19,
fx=1/x-1
fy=2/y-4
fxx=-1/x^2
fxy=0
fyy=-2/y^2
1/x-1=0,	2/y-4=0
(1,1/2)
d(1,1/2)=8>0
fxx(1,1/2)<0
(1,1/2) is a saddle point

23,
fx=-(2x^2+6xy-1)e^(y-x^2)
fy=(3y+x+3)e^(y-x^2)
fxx=2(2x^3+6x^2y-3x-3y)e^(y-x^2)
fxy=-(6yx+2x^2+6x-1)e^(y-x^2)
fyy=(3y+x+6)e^(y-x^2)
-(2x^2+6xy-1)e^(y-x^2)=0,	(3y+x+3)e^(y-x^2)=0
critical point:(-1/6,-17/18)
d(-1/6,-17/18)=4.969>0
fxx=1.173>0
(-1/6,-17/18) is a local minimum

29,
fx=1
fy=1
There is no critical point according to f'(x,y)
x=0,
f(0,y)=y
fy=1
f(0,0)=0
f(0,1)=1
x=1
f(1,y)=1+y
fy=1
f(1,0)=1
f(1,1)=2
y=0
f(x,0)=x
fx=1
f(0,0)=0
f(1,0)=1
y=1
f(x,1)=x+1
fx=1
f(0,1)=1
f(1,1)=2
Therefore, absolute maximum value is 2, absolute minimum value is 0

35
a)
fx=1-2x-y
fy=1-2y-x
1-2x-y=0,	1-2y-x=0
(1/3,1/3)
f(1/3,1/3)=1/3
b)
f'(x,0)=1-2x
1-2x=0,x=1/2
f(1/2,0)=1/4
f(0,0)=0
f(2,0)=-2
max:f(1/2,0)
c)
f(x,2)=-x-x^2-2
f'(x,2)=1-2x
1-2x=0,x=1/2
f(1/2,2)=-11/4
f(0,2)=-2
f(2,2)=-8
max:f(0,2)
f'(0,y)=1-2y
1-2y=0,y=1/2
f(0,1/2)=1/4
f(0,0)=0
f(0,2)=-2
max:f(0,1/2)
f(2,y)=-y-y^2-2
f'(2,y)=1-2y
1-2y=0,y=1/2
f(2,1/2)=-11/4
f(2,0)=-2
f(2,2)=-8
max:f(2,0)
d)
The maximum is f(1/3,1/3)=1/3