Minima and maxima of multivariable equations How do I find the minima and maxima of an equation in the form f(x, y) where x and y are both independent variables? Does the method work in every case? How do I even find out if there is a minimum of maximum?
The equation in question is $f(r, h) = \frac {3r + 3 \sqrt{r^2+h^2}}{rh}$. Are there any minima and maxima? If so, what are they? 
Thanks in advance!
 A: In general, to find the maxima and minima of a given function, you can set the first derivatives, $f_h$, $f_r$ equal to 0 and then check endpoints/places where the function is undefined. Lagrange multipliers are usually applied with a constraint, not to find global maxima and minima of a function.
But for this specific function, we can see that the function does not have an upper bound or lower bound. We see that $f(r,h)$ approaches infinity or negative infinity as r and h go to 0. So intuitively, if we let r and h be arbitrarily smalls number, we see that the function trails off to infinity. If we let r be an arbitrarily small negative number and h be an arbitrarily large positive value, we can see that the numerator will be positive because $\sqrt{r^2+h^2}>3r$ for a sufficiently large h. Thus the function approaches negative infinity in this case because the denominator will be negative, while the numerator is positive. So $f(r,h)$ is unbounded.
We can confirm this graphically:
graph http://www4a.wolframalpha.com/Calculate/MSP/MSP112201f97a25fe092651i000056e28f967f8ai8hg?MSPStoreType=image/gif&s=4&w=300.&h=232.&cdf=MeshControl&cdf=RangeControl
By the way, for Lagrange Multipliers, I recommend PatrickJMT! Check him out: https://www.youtube.com/watch?v=ry9cgNx1QV8.
I hope this helped!
