<s>[INST] How do we minimize $\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right)$? Find the minimum value of the following function, where a and b are real
numbers.
\begin{align} f(x)&=\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right) \end{align}
Note: The solution should not contain any calculus methods.
So I conjugated the parenthesizes two by two, and I got:
\begin{align} f(x)&=\left(x^2-(a+b)^2\right)\left(x^2-(a-b)^2\right) \end{align}
Can somebody help me to take this problem further?
 [/INST] Just another solution in the same spirit as lab bhattacharjee's answer.
$$ f(x)=\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right) $$
$$f(x)=x^4-2\left( a^2+1\right) x^2+(a^2-1)^2$$ Complete the square
$$f(x)=\left(x^2-(a^2+1)\right)^2-(a^2+1)^2+(a^2-1)^2=\left(x^2-(a^2+1)\right)^2-4a^2$$
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