Find $\min$ for$ f(x) = (x + a + b)(x + a - b)(x - a + b)(x - a - b)$ I'm trying to find $minf(x)$ for $f(x) := (x + a + b)(x + a - b)(x - a + b)(x - a - b)$, where $a, b \in \mathbb{R},$ using inequalities. 
For example, i can find $maxf(x)$, using AM-GM ineq:
$$\sqrt[4]{(x + a + b)(x + a - b)(x - a + b)(x - a - b)})^4 \leq \Big (\frac{x + a + b + x + a - b+ x - a + b+ x - a - b}{4}\Big)^4 = $$
$$= x^4.$$
So $maxf(x) = x^4$.
But i don't know how to solve is for $minf(x)$, which i need to find. 
Sure do i can find structure of square difference in $f(x)$ and we can rewrite our equality: 
$$f(x) = (x^2 - (a + b)^2) (x^2 - (b - a)^2).$$
But i don't know what to do next.
UPD: We need to find extremum on $x$ via fixing $a, b$. I understand that $max$ is found wrong way. How can i do it correctly?

Here is another approach:
Denote $t=a+b$ and $s=a-b$, then $f(x)$ becomes to be $(x^2-t^2)(x^2-s^2)$. Henece $f(x)$ is quintic which is very easy to work with, more specifically,
$$f(x)=x^4-(s^2+t^2)x^2+t^2s^2.$$
Now, you can make another substitution, namely $y=x^2$ to obtain a quadratic, which you know how to find a minimum of it. Don't forget to returnt to given variables. 
