I need to find the minimum of
$f(x,y,z) = x^2 + y^2 + z^2$
subject to the constraints
$(x-2)^2 + y^2 + z^2 = 1$
and
$x + z = 3$.
I've got the following equations set up (using Lagrange multipliers):
$$2x = 2(x-2)\lambda+\mu$$
$$2y = 2y\lambda$$
$$2z = 2z\lambda+\mu$$
I suspect the answer is $x=2, y=0, z=1$, but I don't know how to arrive at it.
Is my problem set up correctly? If so how do I solve for $x, y, z, \lambda, \mu$?
Best Answer
From the second equation either $y=0$ or $\lambda=1$. The latter results in an inconsistent system, so set $y=0$ and continue from there. The second constraint allows you to eliminate either $x$ or $z$, and after eliminating $y$ the second constraint becomes a simple quadratic equation in the remaining variable. Since the objective function is the squared distance from the origin, it should be easy to pick out which of the two solutions is minimal.
Of course, like many of the exercises given for practice in the Lagrange multiplier method, this problem can be solved without using the method. Geometrically speaking, you’re being asked to find the nearest point to the origin of the intersection between a sphere and plane. This intersection is a circle. Parameterizing this circle reduces the problem to a straightforward single-variable minimization.