how do i find $\max\{x+z\}$ and $\max\{1+y^2\}$ where ?$x\ge0 $,$ y\ge0$,$ z\ge0$ and$xy+xz+yz=1 $

Question

how compute $\max\{x+z\}$ and $\max\{1+y^2\}$? such that $x$,$y$,$z$ satisfied $$\begin{cases} xy+xz+yz=1 & \\ x\ge0 \\ y\ge0\\ z\ge0\\ \end{cases} $$ i face with this problem when i try solve here

Thanks in advance

Answer 1

We can try Lagrange Multiplier

Alternatively, we know if $A+B+C=\frac\pi2, \sum \tan A\tan B=1$

As $x,y,z\ge0,$ we can choose $A,B,C\in [0,\frac\pi2]$

Now, $1+y^2=1+\tan^2C=\sec^2C$ which tends to $\infty$ as $C\to\infty$

$x+z=\tan A+\tan C$ which tends to $\infty$ as $A$ or $C\to\infty$

Answer 2

$(x+z)$ is unbounded above.Reason: Let $M>0$ be given and let $y=0,z=M,x=\frac{1}{M}$ Then $(x+z)>M$ so it is unbounded above.

$(1+y^2)$ is also unbounded. Reason take $y=M,z=0,x=\frac{1}{M}$ then $(1+y^2)>M$