Obtaining function's extreme values without derivate What is other method to obtain a function min/max value without any use of derivative? 
For example in this function:
$f(x) = 4x + \dfrac{9\pi^2}{x} + \sin(x)$  
My teacher used a method that goes something like this:
$a = 4x$, $\ \ \ $ $b = \dfrac{9\pi^2}{x}$, $\ \ \ \min = 2\sqrt{a \cdot b} - \sin(x)$.
Can anyone tell me name of such method? 

There is a large collection of standard Inequalities that can be used in max/min problems. 
One of the more useful ones is the Arithmetic Mean Geometric Mean Inequality (AM-GM). In the case $n=2$ it says that if $a$ and $b$ are positive then
$$\frac{a+b}{2}\ge \sqrt{ab},$$
with equality only if $a=b$. That  is essentially the fact that you quoted. Note that the inequality can be proved simply by starting from the fact that $(\sqrt{a}-\sqrt{b})^2\ge 0$, and then doing some manipulation.
The General AM-GM says that if the $a_i$ are positive then
$$\frac{a_1+a_2+\cdots+a_n}{n}\ge (a_1a_2\cdots a_n)^{1/n},$$
with equality only when the $a_i$ are all equal.
There are many other inequalities useful for proving max/min results. For example, you might want to look up the Cauchy-Schwarz Inequality. It so happens that the inequality your teacher used can be thought of as using the case $n=2$ of AM-GM or as using the case $n=2$ of C-S, though for larger $n$ they differ.
There are many other "named" inequalities. For a long list, look [here.]  (http://en.wikipedia.org/wiki/List_of_inequalities)
Remark: I strongly recommend the book Maxima and Minima Without Calculus by Ivan Niven.
