For non-negative $x$, $y$, $z$ with $x+y+z=1$, find the minimum and maximum of $\sqrt{x+1}+\sqrt{y+1}+2\sqrt{z+1}$

Question

Let $x$, $y$, $z$ non-negative and $x+y+z=1$. Find the minimum and maximum of $$M=\sqrt{x+1}+\sqrt{y+1}+2\sqrt{z+1}$$

Answer 1

Check the interior then check the boundaries.

Using the method of Lagrange multipliers:

$F(x,y,z,\lambda) = \sqrt {x+1} + \sqrt{y+1} + 2\sqrt {z+1} -\lambda (x+y+z - 1)\\ \frac {\partial F}{\partial x} = \frac {1}{2\sqrt{x+1}} - \lambda = 0\\ \frac {\partial F}{\partial y} = \frac {1}{2\sqrt{y+1}} - \lambda = 0\\ \frac {\partial F}{\partial z} = \frac {1}{1\sqrt{z+1}} - \lambda = 0\\ \frac {\partial F}{\partial \lambda} = x+y+z+-1 = 0$

$2\sqrt {x+1} = 2\sqrt{y+1} = \sqrt{z+1} = \frac{1}{\lambda}\\ x = y\\ z = 4x + 3\\ x+y+z = 6x + 3 = 1\\ x = -\frac{1}{3}$

This gives a suggests a critical point at $(-\frac{1}{3}, -\frac{1}{3}, \frac{5}{3})$

But this isn't in the interior of the region.

But then there is no other critical point in the interior. Now we check the boarders.

Assume $z = 0, x = 1-y$

$M = \sqrt {2-y} + \sqrt {y+1}\\ -\frac {1}{2\sqrt{2-y}} + \frac {1}{2\sqrt{y+1}} = 0\\ 2-y = y+1\\ y = \frac{1}{2}\\ (x,y,z) = (\frac{1}{2}, \frac 12, 0)$

Let $x = 0, y = 1 - z$

$M = \sqrt {2-z} + 2\sqrt {z+1}\\ -\frac {1}{2\sqrt{2-z}} + \frac {1}{\sqrt{z+1}} = 0\\ 8-4z = z+1\\ z = \frac{7}{5}$

Also out off of the region.

$y = 0$ yields the same result.

Finally we check the corners.

We are down to 4 points

$(0,0,1), (0,1,0), (1,0,0), (\frac 12,\frac 12, 0)$

Our objective equals $2+2\sqrt 2, 3+\sqrt 2, 3+\sqrt 2, 2+2\sqrt {\frac{3}{2}}$ respectively

$2+2\sqrt 2, 3+\sqrt 2$ are our minimuma and our maximum.