Lagrange multipliers: find minimum with constraints

Question

Find the minimum of $f(x,y,z) = z$ subject to the constraints $x + y + z = 1$ and $x^2 + y^2 = 1.$

So far, I have $$(0,0,1)=\lambda(1,1,1)+\mu(2x,2y,0)$$

$$0=\lambda+2\mu x \implies 2\mu x=-1 \implies x=-1/2\mu$$

$$0=\lambda+2\mu y \implies 2\mu y=-1 \implies y=-1/2\mu$$

$$1=\lambda$$

I feel like I'm not on the right path as there is no $z$ variable. Any input is appreciated.

Answer 1

The condition you have found is

• $y=x \implies$ from $x^2+y^2=1$ that $2x^2=1 \implies x=\pm \frac{\sqrt 2}2 \quad y=\pm \frac{\sqrt 2}2$

thus from $x+y+z=1$ we have for the extrema points

• $(x,y)=\left(\frac{\sqrt 2}2,\frac{\sqrt 2}2\right) \implies z_{\min}=1-\sqrt 2$

• $(x,y)=\left(-\frac{\sqrt 2}2,-\frac{\sqrt 2}2\right) \implies z_{\max}=1+\sqrt 2$