Lagrange Multiplier Question to find extrema

Question

Find the extrema of f subject to the given conditions. $$f(x,y,z) = x+ y + z\quad \quad x^2-y^2=1 \quad \quad2x+z=1$$

I'm certain to use a lagrange multiplier because it's a surface but am not sure how to do this problem. Please help!

Answer 1

You can simplify your problem by using the equations $$z=1-2x$$ and $$y=\pm\sqrt{x^2-1}$$ then you have the function $$f(x,\pm\sqrt{x^2-1},1-2x)=-x+1\pm\sqrt{x^2-1}$$ in only one variable.

Answer 2

The two constraints define a one-dimensional manifold $M$ in ${\mathbb R}^3$, which happens to be a hyperbola. One could try Lagrange's method for this problem, but would find nothing – for reasons to become clear in a moment.

The equation $x^2-y^2=1$ defines a hyperbolic cylinder that can be parametrized by $$(t,z)\mapsto(\pm\cosh t,\sinh t, z),\qquad-\infty<t<\infty, \ -\infty<z<\infty\ .$$ The equation $z=1-2x$ defines a plane which intersects the cylinder transversally, resulting in a hyperbola. The two branches of this hyperbola are given by the parametric representations $$\eqalign{\gamma_+:\quad &t\mapsto(\cosh t,\sinh t, 1-2\cosh t)\qquad(-\infty<t<\infty),\cr \gamma_-:\quad &t\mapsto(-\cosh t,\sinh t, 1+2\cosh t)\qquad(-\infty<t<\infty)\ .\cr}$$Now we have to set up the pullback of the objective function to the domain ${\mathbb R}$ of these representations. This results in $$\eqalign{\hat f_+(t)&:=f\bigl(\gamma_+(t)\bigr)=1-\cosh t+\sinh t=1-e^{-t},\cr \hat f_-(t)&:=f\bigl(\gamma_-(t)\bigr)=1+\cosh t+\sinh t=1+e^t\ .\cr}$$ Both these pullbacks have nonzero derivative, hence are monotone functions. Furthermore we have $\lim_{t\to-\infty}\hat f_+(t)=-\infty$ and $\lim_{t\to\infty}\hat f_-(t)=\infty$. This shows that $f$ has neither a conditional minimum nor a conditional maximum on $M$.