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So, I came up with another neat math problem (which I actually haven't found the solution to yet). You've got a heart represented by the equation y=2*(2x/3)^(2/3)+/-sqrt(9-(2x/3)^2), and you're trying to find it's height (bottom tip to top curve). Simple enough, right? What I did:
Simply differentiate the equation and solve for which x value made the derivative 0. Easy peasy! I got a cubic equation with coefficients reaching over 1000, and the original equation's approximate answer didn't match the simplified one. I'm betting that something may have gone awry in my calculus as well, so I don't think I can even trust my original equation. After having dealing with the problem for over an hour, I gave up and decided to post it on the TTG fora. Can't wait to see what y'all come up with!

Hmm... well, if you're still working on this problem:

I gave it a try myself, and got a sensible (part-way) answer. Seems to me your approach is a fine one, find out where the derivative is zero. Of course, that won't work for the bottom of the heart (pointy and all that), but you don't need it for that. You just need it for the tops... and there the function doesn't do anything surprising.

My suggestion: replace x with x' = 2*x/3. That way you'll avoid a lot of writing. (One of the biggest lessons I've learning in math: define new variables to make your life easier... you can never* do this too much). After doing the derivative, I ended up with an equation with a few terms, which I then plugged into wolfram alpha, to solve for x'.

So give it another try. What do you get for your derivative?