Find the minimum speed of a particle with trajectory

So here's what I told my professor and he confirmed it as right (though I am still unsure if it was really right):

1. Write the speed formula:

2. Write the derivative of the speed and find where it's equal to 0:

3. Write the second derivative of the speed and find where it's > 0:

So if this is correct, is there a way to simplify this so that I won't need to expand the equation that much? Or am I doing something wrong?