Math
posted by Justin .
Use lagrange multipliers to find the point on the plane x2y 3z14=0 that is closet to the origin?(try and minimize the square of the distance of a point (x,y,z) to the origin subject to the constraint that is on the plane) Help me please!

Read up your textbook or lecture notes on Lagrange multipliers. It is not as difficult as it sounds.
However, you need to master you algebra and basic calculus, which I believe should not be a problem.
To put it in the least words possible, we attempt to optimize (maximize or minimize) a function f(x,y,z) subject to the constraint g(x,y,z).
The suggested objective function is the square of the distance from the origin, which therefore is
f(x,y,z)=x²+y²+z²
The constraint is
g(x,y,z)=x2y+3z14=0
We now introduce a Lagrange multiplier, λ, to form a new function Λ:
Λ(x,y,z)=f(x)λg(x)
Λ(x,y,z)=x²+y²+z²λ(x2y+3z14)
Now apply partial differentiation with respect to each of the variables, and equate result to zero:
∂Λ/∂x=2xλ...(1)
∂Λ/∂y=2y+2λ...(2)
∂Λ/∂z=2z3λ...(3)
Add the original constraint equation:
x2y+3z14=0...(4)
Now solve the system of 4 equations in x,y,z and λ and voilĂ !
x=1, y=2, z=3, L=2.
So the distance is
D=√(1²+(2)²+3²)
=√14
How can we tell if this is correct?
This is not too difficult... in this particular case.
The shortest distance from a point (origin 0,0,0) to a plane is the perpendicular distance, given by the wellknown formula:
Dmin=(ax0+by0+cz0+d)/√(a²+b*sup2;+c²)
=(00+014)/√(1²+(2)²+3²)
=14/√(14)
=√(14)
and the square of the minimum distance
Dmin²=14
which checks with our Lagrange multiplier answer.
Respond to this Question
Similar Questions

Calculus
"Using Lagrange multipliers, find the minimum value of f(x,y) = x^2 + y subject to the constraint x^2  y^2 = 1." Any help would be appreciated! 
Calculus
"Using Lagrange multipliers, find the maximum value of f(x,y) = x + 3y + 5z subject to the constraint x^2 + y^2 + z^2 = 1." Any help would be appreciated! 
calculus
Use Lagrange multipliers to find the max/min values of the function f(x,y)=xy subject to the constraint: x^2/8+y^2/2 =1 Pleasssse help me with this!! 
cal3 please help!
Use Lagrange multipliers to find the max/min values of the function f(x,y)=xy subject to the constraint: x^2/8+y^2/2 =1 so I compare the gradient vectors of both f(x,y) and the constraint: <y,x>=L<x/4,Ly> resulting in y=Lx/4 … 
:( please help answer cal problem
Use lagrange multipliers to find the max and min values of the func f(x,y)=xy subject to the constraint 1=(x^2)/8 + (y^2)/2 I know how to set up.. i got y=Lx/4 and x=Ly now im lost!!! Please help with detail because im an idiot. 
Physics
A system consists of the following masses loacted in the xy plane: 5.0 kg at the point (0,5), 7.5 kg at the point (3,8) and 6.5 at the point (3, 6). Find the distance from the origin to the center of mass. 
Physics
A system consists of the following masses loacted in the xy plane: 5.0 kg at the point (0,5), 7.5 kg at the point (3,8) and 6.5 at the point (3, 6). Find the distance from the origin to the center of mass. 
Calculus III
Hi, this question is about Langrange multipliers. Given f(x,y) = y^2  x^2, subject to the constraint g(x,y) = 0.25x^2 + y^2 = 1, find the max and mins. So I found the partial derivatives for both f(x,y) and g(x,y): fx = 2x fy = 2y … 
Calculus III
Hi, this question is about Langrange multipliers. Given f(x,y) = y^2  x^2, subject to the constraint g(x,y) = 0.25x^2 + y^2 = 1, find the max and mins. So I found the partial derivatives for both f(x,y) and g(x,y): fx = 2x fy = 2y … 
Calculus III
Hi, this question is about Langrange multipliers. Given f(x,y) = y^2  x^2, subject to the constraint g(x,y) = 0.25x^2 + y^2 = 1, find the max and mins. So I found the partial derivatives for both f(x,y) and g(x,y): fx = 2x fy = 2y …