Let's start with saying that you were able to take the derivatives correctly.
If I understood it correctly you wanted to find the x for where the first derivative equals 0. Here is where you started typing down things that didn't make much sense. You say you filled in the -1 and the +1 for...
This might not be the answer you may have hoped for. I wonder what sites told you that all conservative fields can produce an harmonic motion.
As a counter example:
We have the potential function f=1x
then the vector field will become F=\nabla f=1
Good luck getting an harmonic...
Your answer is correct.
I was just not completely happy with how you reached the K.E. formula. But it seems you understand what is going on so I will leave it at that.
edit: E_{0}=mgL(1-cos(\theta_{0})) is not a formula for kinetic energy as it comes from mg\Delta h, which is for potential...
Correct. But the thing that bothers me is:
That formula is derived from the height the pendulum reaches. It has nothing to do with the kinetic energy at \theta=0. Besides that in a frictionless environment the potential energy at maximum angle is the same as the kinetic energy at the lowest...
When do you think the ball will fall from the cone? Try it out with a glass and a ball.
You will see that once the center of mass of the ball goes 'further' than the point it is resting on, it will fall out.
Now you'll just have to figure out at what angle that will be.
Tips:
1. How much is A in front of B? (90 deg)
How big is the phase difference between A and B just because wave A has to travel longer?
2. For both 450 nm and 580 nm a double-slit interference will occur. What distance for m=4 at 450 nm and what distance for the 580 nm wave?