# Math Help - confusing lagrange multiplier question

1. ## confusing lagrange multiplier question

f(x,y,z)=xy+z^3
subject to x^2+y^2+z^2=1
Find all the absolute maximum and minimum.

I tried to solve the simutaneous equations, and I ended up with 14 critical points!
So I doubt if I did right or wrong.
Can anybody please help me and see how many of them? What are they? I solved that x=o, plus or minus square root 2 on 2, plus or minus 2/3
y=0, plus or minus square root 2 on 2, plus or minus 2/3
z=0, plus or minus 1, plus or minus 1/3
Which rearrange them, I end up with 14 critical points. Am I right? Thanks a lot.

2. The equations are

$y=2 x \lambda$
$x=2 y \lambda$
$3 z^2=2 z \lambda$
$x^2+y^2+z^2=1$

Right?
If $x = 0$, so is $y = 0$. Then from the last equation $z = +1$ or $z = -1$. These are 2 points.
If $z = 0$, then you get $x = \sqrt{2}/2$ or $x = -\sqrt{2}/2$ and $y = \sqrt{2}/2$ or $y = -\sqrt{2}/2$. All four combinations are valid, so you get another 4 points.

The last option is if none of them are 0. Then \lambda is non-zero also because if it was, then we'd get one of $x,y,z = 0$. If you solve this you should get 4 solutions where $x = 2/3$, $y = 2/3$, $z = 1/3$ (where some of the signs change, i.e. $z = 1/3$, $x = 2/3$, $y = -2/3$)

This is a grand total of 10 solutions. The way you did it is incorrect.