When finding relative minimum/maximum, what is the point of using the second derivative test? Why use the second derivative test over the first derivative test when finding maxima and minima if it's uncertain what $f''(x) = 0$ is? Why not just always use the first derivative since we need to take the first derivate either way?
Here's what I tell my students:


The first derivative test is usually harder to use, but will always give you an answer (at least for reasonable functions);
The second derivative test is usually easier to use, but it may not be applicable (if it is not a stationary point) and it may be inconclusive.


Why do I say that? To use the second derivative test, you just need to plug in a number and evaluate. That's straightforward. To use the first derivative test, you need to analyse the sign of the first derivative on a neighborhood of the point and figure out whether it changes signs as it goes through the point. This can be harder to do than just plugging in a single point.

As you say, you always already have the first derivative handy. But you may also need to have the second derivative handy: for example, if you are doing a full analysis of the function, looking for intervals in which the function is increasing and decreasing, and intervals in which the function is concave up and concave down, you will also have the second derivative handy. 

It's a tradeoff, then:


For the First Derivative Test, you can always use it (stationary points and points where the first derivative does not exist), will almost always give you an answer either way (detecting critical points that are not relative extremes), and does not require any further derivatives to be calculated. But it requires a sign analysis, which may not be trivial (or possible). 
For the Second Derivative, you can't always use it, and even if you can use it, it may be inconclusive and it will never detect critical points that are not relative extremes, and it requires you to calculate another derivative. But it is usually much easier to actually use, as it only requires figuring out the sign at a single point. 


But more generally: it's always good to have more than one way to do something, as different ways may provide strengths and weaknesses that the other methods don't have, or provide a different kind of insight.
Because a zero of the derivative has no sign ! Hence it cannot tell a minimum from a maximum.

And what about $f(x)=x^3$ ?