F(x,y,z)=4x^2+y^2+5z^2. Use Lagrange Multipliers to find the point on the plane 2x+3y+4z=12 at which F(x,y,z) is the least value.
Δf(x,y,z)= 8xi+2yj+10zk
γΔg(x,y,z)= 2λ+3λ+4λ
8x=2λ
3y=3λ
10z=4λ
Im stuck
might be wrong, but here goes:
g= lambda
since 8x = 2g
that means that 8 = 2g/x
which means that 12 = 6g/2x
so 6g/2x = 2x +3y + 4z
6g = 2x(2x + 3y + 4z)
6g = 1/2g(1/2g + 3g + 4z)
12g^2 -3.5g = 4z
10z = 4g
z = 2/5g
8/5g + 3.5g =12g^2
1.6g + 3.5g = 5.1g
5.1g/g
5.1 = 12g
go from there.. i think