How to find equality of maximum or minimum value of the given expression? I understood that if expression and condition are symmetric, then we may assume that all of the variables are equal. But how to know at least one of the expression and condition is not symmetric?
For example, $x,y,z\in\mathbb{R^{+}}$ and $3x^2+4y^2+5z^2=2xyz$ are given. Then how to find equality the minimum value of this expression $3x+2y+z$?
By AM-GM $$3x+2y+z=\frac{(3x^2+4y^2+5z^2)(3x+2y+z)}{2xyz}\geq$$ $$\geq\frac{12\sqrt[12]{(x^2)^3(y^2)^4(z^2)^5}\cdot6\sqrt[6]{x^3y^2z}}{2xyz}=36.$$ The equality occurs for $x=y=z$ and $3x^2+4y^2+5z^2=2xyz,$ which says that we got a minimal value.
Also, you can use the following way.
Let $f(x,y,z,\lambda)=3x+2y+z+\lambda(3x^2+4y^2+5z^2-2xyz).$
Thus, in the minimum point we have $$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=\frac{\partial f}{\partial \lambda}=0$$ and you'll get the system, which gives that $(6,6,6)$ is a critical point and by using second partial derivatives we can get that it's a minimum point.
I think, it's better to look for the first way before.