1. ## Multi-Variables Calculus [Max,Min,Lag.Multipliers]

Hey there,

I realy need some help in the following question:
Find absolute maximum and minimum of the function:
$f(x,y,z)=e^{-2x-3y-5z}$ in the set:
${(x,y,z) | x^{2}+y^{2}+z<=1,x+y+z>=1 }$ .
{Does the fact the the function $\phi (t) =e^{t}$ is monotonic can help? } ...

Well, we know that the only place where the partial deriatives are zero is (0,0,0)... But it doesn't tell us anything about absolute min/max... I'll be delighted to get some help on how to continue...

Thanks!

2. Originally Posted by WannaBe
Hey there,

I realy need some help in the following question:
Find absolute maximum and minimum of the function:
$f(x,y,z)=e^{-2x-3y-5z}$ in the set:
${(x,y,z) | x^{2}+y^{2}+z<=1,x+y+z>=1 }$ .
{Does the fact the the function $\phi (t) =e^{t}$ is monotonic can help? } ...

Well, we know that the only place where the partial deriatives are zero is (0,0,0)... But it doesn't tell us anything about absolute min/max... I'll be delighted to get some help on how to continue...

Thanks!
You'll notice that

$\left[\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right] = \left[-2e^{-2x - 3y - 5z}, -3e^{-2x - 3y - 5z}, -5e^{-2x - 3y - 5z}\right]$.

Since the exponential function is always $>0$, that means there are not any stationary points in this region.

So the absolute minimum and maximum must be somewhere on the boundaries.

3. Yep...You're right...I've managed to get to this point too (my statement about (0,0,0) is wrong indeed...) ... But my problem is I can't figure out how to find the absolute min/max and how to prove they're absolute min/max ....
The boundaries of this region are
${(x,y,z) | x^{2}+y^{2}+z=1,x+y+z=1 }$ ...How can I find the absolute min/max in this region?

Hope you'll be able to help me

Thanks a lot!