14.8  :

5,
fx=2x	fy=2y
grad(f)=<2x,2y>
gx=2	gy=3
grad(g)=<2,3>
<2x,2y>=λ<2,3>
{2x=2λ,2y=3λ,2x+3y=6}
{x=12/13,y=18/13,λ=12/13}
f(12/13,18/13)=36/13
Minimum: 36/13

7,
fx=y	fy=x
grad(f)=<y,x>
gx=8x	gy=18y
grad(g)=<8x,18y>
<8x,18y>=λ<y,x>
{8x=λy,18y=λx,4x^2+9y^2=32}
{-2,-4/3,12}{2,4/3,12}
f(-2,-4/3)=8/3
f(2,4/3)=8/3
Maximum:8/3

9,
fx=2x	fy=2y
grad(f)=<2x,2y>
gx=4x^3	gy=4y^3
grad(g)=<4x^3,4y^3>
λ<2x,2y>=<4x^3,4y^3>
{λ2x=4x^3,λ2y=4y^3,x^4+y^4=1}
{0,-1,2}{1,0,2}{(2^(3/4))/2,2^(3/4))/2}
f(0,-1)=1
Minimum:1 
f((2^(3/4))/2,2^(3/4))/2)=sqrt2
Maximum:2

11,
fx=3	fy=2	fz=4
grad(f)=<3,2,4>
gx=2x,	gy=4y,	gz=12z
grad(g)=<2x,4y,12z>
λ<3,2,4>=<2x,4y,12z>
{3λ=2x,2λ=4y,4λ=12z,x^2+2y^2+6z^2=1}
{+-(3*sqrt123)/41,+-(sqrt123)/41,+-(2*sqrt123)/123,+-(2*sqrt123)/41}
f((3*sqrt123)/41,(sqrt123)/41,(2*sqrt123)/123)=sqrt(123)/3
Maximum:sqrt(123)/3
f(-(3*sqrt123)/41,-(sqrt123)/41,-(2*sqrt123)/123)=-sqrt(123)/3
Mimimum:-sqrt(123)/3

13,
fx=y	fy=x	fz=2
grad(f)=<y,x,2>
gx=2x	gy=2y	gz=2z
grad(g)=<2x,2y,2z>
λ<y,x,2>=<2x,2y,2z>
{λy=2x,λx=2y,λ2=2z,x^2+y^2+z^2=36}
{0,0,-6,-6}{-4,4,-2,-2}{4,-4,-2,-2}{-4.-4,2,2}{4,4,2,2}{0,0,6,6}
f(0,0,-6)=-12,f(-4,4,-2)=-20,f(4,-4,-2)=-20,f(-4.-4,2)=20,f(4,4,2)=20,f(0,0,6)=12
Maximum:20,
Minimum:-20

15
fx=y+z	fy=x	fz=x
grad(f)=<y+z,x,x>
gx=2x,	gy=2y,	gz=2z
grad(g)=<2x,2y,2z>
λ<y+z,x,x>=<2x,2y,2z>
{λ(y+z)=2x,λx=2y,λx=2z,x^2+y^2+z^2=4}
{-sqrt2,1,1,-sqrt2}{sqrt2,-1,-1,-sqrt2}{-sqrt2,-1,-1,sqrt2}{sqrt2,1,1,sqrt2}
f(-sqrt2,1,1)=-2sqrt2
f(sqrt2,-1,-1)=-2sqrt2
f(-sqrt2,-1,-1)=2sqrt2
f(sqrt2,1,1)=2sqrt2
Maximum:2sqrt2
Minimum:-2sqrt2