locating a point closest to the origin using Lagrange multipliers?
Here is the question...
Locate the point on the line which is the intersection of the planes and which is closest to the origin. Can someone solve this and please tell me how to do it using Lagrange multipliers? Thanks in advance.
Here is the question...
Locate the point on the line which is the intersection of the planes and which is closest to the origin. Can someone solve this and please tell me how to do it using Lagrange multipliers? Thanks in advance.
Let f(x, y, z) = x^2 + y^2 + z^2, g(x, y, z) = y + 2z, h(x, y, z) = x + z. Then you want to minimize f under the constraints g = 12 and h = 6. To do so, solve the system of five equations generated by grad(f) = lambda*grad(g) + mu*grad(h), g = 12, h = 6.
Let f(x, y, z) = x^2 + y^2 + z^2, g(x, y, z) = y + 2z, h(x, y, z) = x + z. Then you want to minimize f under the constraints g = 12 and h = 6. To do so, solve the system of five equations generated by grad(f) = lambda*grad(g) + mu*grad(h), g = 12, h = 6.
Hi, yea I did that part. For the 5 equation I got the following...
1)
2)
3)
4)
5)
I can't figure out to solve for the variables though. Can you help me out with this? Thanks.
Hi, yea I did that part. For the 5 equation I got the following...
1)
2)
3)
4)
5)
I can't figure out to solve for the variables though. Can you help me out with this? Thanks.
Since and , 2z= 2(2x)+ 2(2y) so z= 2x+ 2y. Putting that into y+ 2z= 12 gives y+ 2(2x+2y)= 4x+ 5y= 12.
Putting z= 2x+ 2y into x+ z= 6 gives x+ (2x+2y)= 3x+ 2y= 6.