Difference between a < x < b and x < a, x > b when finding increasing or decreasing function? I have been looking at how to find increasing and decreasing functions using differentiation. To give an example of what I am looking at a typical question in this area of mathematics would consist of: 
Is f(x) = 3x3 - 2x an increasing or decreasing function when x < a and x > b?
Would there be any difference if, instead of being given:
x < a and x > b 
I was given a < x < b? 
Would this change anything, in terms of whether the function is now increasing or decreasing?
I do know that if dy/dx > 0 then the function is increasing, and so on and so forth, however I just need clarification on the above.
 A: $$a<x<b$$ means "$a<x$ and $x<b$", which is different from "$a>x$ and $x>b$".  The former only gives values $x$ when $a<b$, while the latter only gives values $x$ when $a>b$.
Further, in the context of this specific question, the word "and" should be considered to be "or".  That is, this question has two parts: one for $x<a$, and the other for $x>b$.
A: we have $$f(x)=3x^3-2x$$ thus we get $$f'(x)=9x^2-2$$ and we can solve $$f'(x)>0$$ if $$x^2>\frac{2}{3}$$ and we get $$|x|>\frac{\sqrt{2}}{3}$$ can you finish?
and for $$X\geq 0$$ we get $$x\geq \frac{\sqrt{2}}{3}$$ or for $$x<0$$ we find $$x<-\frac{\sqrt{2}}{3}$$
A: Assuming $a \lt b$, which would be normal for questions like this, $a \lt x \lt b$ lets $x$ range between $a$ and $b$, while $x\lt a$ and $x \gt b$ is the rest of the real line.  In particular, a cubic with positive leading coefficient can increase from very negative values to a local maximum, decrease for a while to a local minimum, then increase without bound.  If $a,b$ are chosen as the extrema, the function will be decreasing when $a \lt x \lt b$ and increasing when $x \lt a$ and when $x \gt b$.  When you see "$x \lt a$ and $x \gt b$" you need to figure out from context whether you are being given two conditions and asked for two answers, one that applies when $x \lt a$ and another (which might be different) that applies when $x \gt b$ or the two conditions are supposed to be true simultaneously.  If $a \lt b$ in this example, that would be never, which would be an indication that you were supposed to consider the conditions separately.
