[Edited]
The sentence " the minimum value of a product is the product of the minimum values of its factors" seems comformable to common sense.
Is it correct however? What qualifications would it require?
[ Previous version, involving calculation mistakes]
The sentence " the minimum value of a product is the product of the minimum values of its factors" seems comformable to common sense.
However, while thinking about the question ( asked recently) : " what is the set of values the expression $2x^2 - 2x$ can take while $x$ ranges over $[0,2]$?" , the above intuitive principle induced me into error.
The original expression can be factored : $2x^2 - 2x$ = $(2x) (x-1)$
If $x$ ranges over $[0,2]$, then the minimum value of $2x$ is $0$ and the minimum value of $x-1$ is $-1$. .
So, apparently, the minimum value of the product should be $0$ ( that is, the product of the minimum values of the factors).
But in fact, the minimum value is $-\frac 12$ ( as a graph shows).
Hence my question : how to show that the alledged intuitive principle is wrong? or what qualifications would it require?