Question

Use Lagrange multipliers to find minimum and maximum values of the function subject to the constraints given:

f(x,y,z)=(x^2)*y+z subject to x^2+y^2=1, z=y

Answer 1

where are you stuck?

Answer 2

I can't solve the equations after differentiating the components.

dL/dx=2xy+2xr=0
dL/dy=x^2+2yr=0
dL/dz=x^2+1+2zr=0

so from the first equation:

2xy+2xr=0
x(y+r)=0 ; therefore x=0, y=-r

second equation:

x^2+2yr=0
0+2yr=0
2(-r)r=0 ; therefore r=0

third equation:

x^2+1+2zr=0
0+1+2z(0)=0
1=0 <--- I guess I'm stuck here

Answer 3

you have 2 constraints...are you using two Lagrange multipliers?

Answer 4

One constraint function:

z=y or g(y,z)=y-z

and one function that needs to be optimized:

f(x,y,z)=(x^2)*y+z subject to x^2+y^2=1