The Second derivative test for Concavity

The first derivative gives the equation of the gradient. For a concave UP (aka MAXIMUM) the value of the gradient is decreasing, i.e. it goes from +ve, through zero, to -ve , as the tangent travels round the curve from left to right.
see the orange section of the graph which demonstrates this in the green domain (and vice versa for concave DOWN). The second derivative gives the rate of change of the gradient. If the second derivative is -ve, if means the gradient is decreasing, hence the curve is concave UP.
Differentiation of a quadratic gives the equation of the gradient as a linear function, and the second derivative as a constant. (A non-zero constant implies a single maximum or minimum which is what you expect for a parabola)

You're assuming that >0 means a constant value over the entire range of x and that's not the case.

Let's take y=x^3, so the second derivative is y'' = 6x. For any x>0, y'' is also greater than zero and the function is concave up. For any x<0, the opposite is true.

Thanx Everybody, My doubt is clear now. Just like First derivative Test tells us the behaviour of the function whether it is increasing or decreasing. The Second Derivative Test tells us the behaviour of the slope of the function whether it is increasing or decreasing.