# Math

Use lagrange multipliers to find the point on the plane x-2y 3z-14=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!

1. 👍 0
2. 👎 0
3. 👁 199
1. 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)=x-2y+3z-14=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+z²-λ(x-2y+3z-14)

Now apply partial differentiation with respect to each of the variables, and equate result to zero:
∂Λ/∂x=2x-λ...(1)
∂Λ/∂y=2y+2λ...(2)
∂Λ/∂z=2z-3λ...(3)
x-2y+3z-14=0...(4)

Now solve the system of 4 equations in x,y,z and λ and voilà!

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 well-known formula:
Dmin=(ax0+by0+cz0+d)/√(a²+b*sup2;+c²)
=(0-0+0-14)/√(1²+(-2)²+3²)
=-14/√(14)
=-√(14)
and the square of the minimum distance
Dmin²=14
which checks with our Lagrange multiplier answer.

1. 👍 0
2. 👎 0
posted by MathMate

## Similar Questions

1. ### 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!

asked by Anonymous on April 4, 2011
2. ### 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!

asked by Anonymous on April 4, 2011
3. ### 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 =

asked by Justin on December 11, 2016
4. ### 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 =

asked by Justin on December 9, 2016
5. ### 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 =

asked by Justin on December 10, 2016
6. ### 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!!

asked by Lucy on February 27, 2012
7. ### calculus

Use Lagrange multipliers to find the maximum and minimum values of f(x,y)=x^2y+3y^2-y subject to the constraintx^2+y^2less than or equal to 10

asked by RAJ on October 17, 2014

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.

asked by Lucy on February 28, 2012