Q:

The second derivative test tells you the concavity of a graph but what's the point if you can tell the concavity by the leading coefficient?

For example, in $-x^3$, I know it's concave down. The second derivative would tell me that as well. But, please explain why it's needed.

A:

You can't tell the concavity of a graph from the leading coefficient. First of all, only polynomials have a leading coefficient, and even for such functions, this does not tell you about its concavity. For example, $f(x) = x^3 + 3x^2$ has a positive leading coefficient, but it has second derivative $6x + 6$, so it is concave down for $x < - 1$ and concave up for $x > -1$.
Added Later: Simpler still, the function $f(x) = -x^3$ which you claim is concave down is not. It has second derivative $-6x$, so it is concave up for $x < 0$ and concave down for $x > 0$.

A:

The second derivative test is about determining if a critical point is a local extremum, not about telling the concavity.

