Graphing and differentiation I am having trouble making sense of how to figure out what a graph looks like just by knowing the critical numbers and the intervals the function is increasing and when it is decreasing. For example I have $(-\infty, -1), (0,1)$ as negative and $(-1,0), (1,+\infty)$ as positive.
I know that the it can only change on critical numbers so that means if I have increase, critical number and then decreasing then I have a graph that is going up and then down and the critical number is a minimum. If I have decrease, critical number and then an increase I have a minimum.
The problem I ma having is determing how to view these. If I start from the left of the function I get concave up (minimums) with decrease, critical number, increase.
If I do the same with an increase, critical number decrease by starting from the right side I get a concave down (max).
How do I know which is correct when working with just this information? It seems to depend entirely on which side I start on.
 A: The sign of your function seems to change at $-1$, $0$ and $1$ rather than the direction.
You have not specified much more about your function.  If it is continuous then it will have at least one local maximum in $(-1,0)$ and the direction will change there, and similarly at least one local minimum in $(0,1)$.  This will not let you conclude much about where it will be concave or convex (apart from being concave up close to a local minimum etc.).
I would have thought each of the red and blue functions below would meet your specifications:
 
If I have misunderstood and it is the direction which changes then you still cannot conclude anything about concavity and convexity, though there will be local minima at $-1$ and $1$ (at least one of which will be a global minimum) and a local maximum at $0$ 

A: Take a look at: Link-1 for a good explanation of the concept.
Take a look at the graph below.
Your critical numbers are -1, 0, 1 are shown.
To determine your local min/max use the rule shown below.
Second derivative is = ${f}''(x)=12 x ^{2} - 4 $
Second derivative sign at critical points:
at -1 --> (+) --> local min.
at  0 --> (-) --> local max.
at  1 --> (+) --> local max.

