Convex Optimization Problem (example)

Your idea is good, since checking via Hessian is a simple technique in case it's easy to calculate it. However, with more complex functions the Hessian can be hard to compute, so it's a good practice to prove convexity using "non derivative" methods.

In your case, ask yourself the following (works for the target function as well as the constraints): What are the atoms ("building blocks") of the function? Do you see a simple decomposition to the function? For example, is the function a summation of convex functions? Recall that summing convex functions maintains convexity.