The Lagrangian is
[tex]L(x,y,z,\lambda_1,\lambda_2)=z+\lambda_1(x^2+y^2-z^2)+\lambda_2(x+y+z-24)[/tex]
with partial derivatives (set equal to zero) yielding
[tex]\begin{cases}L_x=2x\lambda_1+\lambda_2=0\\L_y=2y\lambda_1+\lambda_2=0\\L_z=1-2z\lambda_1+\lambda_2=0\\L_{\lambda_1}=x^2+y^2-z^2=0\\L_{\lambda_2}=x+y+z-24=0\end{cases}[/tex]
Subtracting the second equation from the first gives
[tex](2x\lambda_1+\lambda_2)-(2y\lambda_1+\lambda_2)=2\lambda_1(x-y)=0\implies x=y[/tex]
So in the fourth and fifth equations, we have
[tex]\begin{cases}2x^2=z^2\\2x+z=24\end{cases}\implies x=24\pm12\sqrt2,z=-24\mp24\sqrt2[/tex]
There are then two critical points for [tex]f(x,y,z)=z[/tex] at [tex](24+12\sqrt2,24+12\sqrt2,-24-24\sqrt2)[/tex] and [tex](24-12\sqrt2,24-12\sqrt2,-24+24\sqrt2)[/tex].
Clearly, [tex]f(x,y,z)=z[/tex] attains a local minimum at the first point of [tex]-24-24\sqrt2[/tex], and a local maximum at the second point of [tex]-24+24\sqrt2[/tex].
