Why do we HAVE to test the first and second derivative to determine the nature of a stationary point? Classically, students are taught two ways of determining the nature of a stationary point of some curve $y=f(x)$.
The first is to use a table of values and to test say $x=0.9$, $x=1$ and $x=1.1$ if the stationary point is at $x=1$. These values are input into the gradient function to determine the slopes before and after the stationary point.
The second is to substitute $x=1$ directly into the second derivative and examining the sign of the resultant number.
My question is, why is it necessary to use these methods? Why can we not simply substitute these values into our original function to observe the behaviour of the curve around that stationary point, and thus the nature of it?

Your first approach, using values, can be fooled if there is structure to the curve at a smaller scale.  If I define $f(x)=(x-0.99)^2(x-1.01)^2$ it has a stationary point at $x=1$.  Looking at the values at $x=0.9, 1.1$ one would conclude that $x=1$ is a local minimum, but it is not.  The second derivative test will get it right.  
I'm not sure what you mean by "substitute these values into our original function to observe the behaviour of the curve".  What would you do in this case?
