How do I minimize the distance between the origin and a sphere using Lagrange multipliers?

Question

The question is: Using Lagrange multipliers, which point on the sphere, $x^2+y^2+(z-4)^2=1$, is closest to the origin, $(0,0,0)$?

I decided to minimize the distance function squared: $$f(x,y,z)=x^2+y^2+z^2$$

I determined the constraint to be $$g(x,y,z)=x^2+y^2+(z-4)^2-1=0$$ I found that $$\nabla f = 2x\hat{i} + 2y\hat{j}+ 2z\hat{k}$$ $$\nabla g = 2x\hat{i} + 2y\hat{j}+ 2(z-4)\hat{k}$$ Plugging these into $\nabla f=\lambda\nabla g$, I found these relationships: $$2x=2\lambda x$$ $$2y=2\lambda y$$ $$2z=2\lambda (z-4)$$

Simplifying these relationships, I get $$1=\lambda$$ $$1=\lambda$$ $$z=\lambda (z-4)$$ These relationships don't work out, since plugging $\lambda =1$ into $z=\lambda z-4$ yields $0=-4$ which simply isn't true.

I know conceptually that I have a sphere of radius $1$ that is $4$ units above the $z$-axis which means that the closest distance should be at a point where $z=3$. I have also entered “minimize $f(x,y,z)=x^{2}+y^{2}+z^{2}$ with constraint $x^{2}+y^{2}+(z-4)^{2}-1=0$” into WolframAlpha, and this proved my conceptualization. However, I can’t seem to figure out how to mathematically prove this. Any help is appreciated. Thank you!

[Edit] Simplifying these relationships, I get

$$1=\lambda \text{ or } x = 0$$ $$1=\lambda \text{ or } y = 0$$ $$z=\lambda (z-4)$$

Answer 1

Your simplifications are wrong. From the equality $2x=2\lambda x$, what you get is that $\lambda=1$ or that $x=0$. For the same reason, $\lambda=1$ or $y=0$.

Answer 2

If $\lambda=1$, the last relationship would yield $0=-4$, so we will deduce $x=0$ and $y=0$ instead. Since we don't know what lambda is, we will have to use the constraint to solve for z.
Rearraging the constraint to find z, we get $$z=\sqrt{1-x^2-y^2}+4$$ Plugging in $x=0$ and $y=0$, we get $$z=\pm\sqrt{1}+4$$ $$z=3,5$$ So, the possible solutions are $$(0,0,3) \text{ or } (0,0,5)$$ $(0,0,3)$ is correct since it is closest to the origin.

Answer 3

You could also have avoided this confusion - indeed, Lagrange multipliers altogether - by noting that minimising $x^2+y^2+z^2$ subject to minimising $x^2+y^2+z^2-8z+15=0$ is equivalent to minimising $8z-15$ (or, indeed, $z$) subject to that constraint. Since $$x^2+y^2+(z-4)^2=1\implies (z-4)^2\le1\implies z\ge 3,$$the solution must use $z=3$ so $x=y=0$.

Answer 4

Alternative solution:

In spherical coordinates around the origin $(0,0,4)$, you minimize

$$\sqrt{\cos^2\phi\sin^2\theta+\sin^2\phi\sin^2\theta+(\cos\theta+4)^2}=\sqrt{8\cos\theta+17}.$$

The minimum is $3$ (and the maximum $5$).