consider this equation, $$y=x(400-x)$$the second derivative of this equation is $$y''=-2$$ As far as I know, a negative sign in the second derivative indicates the curve will concave down. As it is a constant I think it says that the curve concaves down all the time. Which means the tangent line will always lie above the function's graph. 
QUESTION 1: How can second derivative define concavity? Surely it's not like mathematicians came up with this out of nowhere.
QUESTION 2: What is the significance of knowing where the graph concaves up or down? There may be many but I just to know the absolute basic ones.
EDIT: I messed up badly with concavity last time. But wikipedia made my views clear about concavity. So I edited my questions.
Best Answer
Negative second order derivative means the first order derivative is decreasing (doing down).
A negative first order derivative means the curve is decreasing (going down)
In our case, the second derivative is negative, which means the first order derivative is decreasing - it is still positive for $x<200$; as long as first order derivative is positive, the curve is going up.