Why is convexity more important than quasi-convexity in optimization? In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have globally optimal solutions, so one can use methods such as gradient (steepest) descent to find such solutions, but I am not convinced.

For example, the function $|x|^{0.001}$  is not convex (see the yellow-shaded area in the picture below) but it has a single global minimizer.

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So why is convexity so important in optimization? Why is e.g. quasiconvexity often not enough?
There are many reasons why convexity is more important than quasi-convexity in optimization theory.  I'd like to mention one that the other answers so far haven't covered in detail.  It is related to Rahul Narain's comment that the class of quasi-convex functions is not closed under addition.

Duality theory makes heavy use of optimizing functions of the form $f+L$ over all linear functions $L$.  If a function $f$ is convex, then for any linear $L$ the function $f+L$ is convex, and hence quasi-convex.  I recommend proving the converse as an exercise: $f+L$ is quasi-convex for all linear functions $L$ if and only if $f$ is convex.

Thus, for every quasi-convex but non-convex function $f$ there is a linear function $L$ such that $f+L$ is not quasi-convex.  I encourage you to construct an example of a quasi-convex function $f$ and a linear function $L$ such that $f+L$ has local minima which are not global minima.  

Thus, in some sense convex functions are the class of functions for which the techniques used in duality theory apply.
I'm not an expert, but just from eyeballing your function you can see that gradient descent would not converge to the global minimizer.  Gradient descent looks for places where the gradient (which is just the derivative for 1-dimensional functions like the one you provide) is zero.  In a convex function, as you approach the minimum, the gradient approaches zero (and vice versa).  But you can see that, as you approach the minimum in your function, the derivative is getting further and further from zero.

If you try to do gradient descent on that function, you'll actually move further and further away from the minimum, since the gradient is minimized as you approach (positive or negative) infinity.

So at least one reason convexity is so important in optimization is that the global minimum is also the unique critical point (place where the gradient is zero), which allows you to search for one by searching for the other.  In some sense, you pick a point, look around a little bit, and move in the direction that makes your gradient smaller.  Repeat this enough and you'll get to a really small gradient, which corresponds to being close to the minimum.