There is no general method except by checking where the derivitive is 0 (or does not exist). For some functions, there are other ways. For example we can always find minima and maxima for quadratic functions by completing the square.
But there is also possibility to estimate. If you solve some elementary function, for example: [tex]f(x)=x^2+3x+2[/tex] You can transform it to form: [tex]f(x)+\frac{1}{4}=\left(x+\frac{3}{2}\right)^2[/tex] So now you are able to find a minimum: [tex]\min_{x\in\mathbb{R}}f(x)=-\frac{1}{4}[/tex]
Yes, but my function is far too complex/tedious to do either way. An expression for the min and max has been found though proving it is too difficult for me.
Well if you are working with quadratics a lot you should know that the min/max is going to be at [tex] \left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)[/tex]