Here's an interesting problem (interesting to me, at least) that my professor gave me last year (outside of class...it had nearly nothing to do with the subject we were studying). It's of two parts: The first part is fairly simple. Suppose you have the graph [tex]f(x)=x^2[/tex]. What is the radius of the largest circle that you can "drop" into the parabola so that one point at the bottom of the circle touches the bottom tip of the parabola. So, said again for redundancy's sake, what is the radius of the largest circle that can fit inside a standard parabola and still touch the bottom. The second part took me more time, but that's because I kept going down the same dead-end road over and over again. Basically, it's the same as the above, except using [tex]f(x)=x^4[/tex] instead of the standard parabola. So, once again for redundancy's sake, what is the radius of the largest circle that can fit inside the graph [tex]f(x)=x^4[/tex] and still touch the bottom-most point. If I remember correctly, this one will touch at more than one point. I'm curious to see how many ways there are to do this. I know of two ways already: the method that I used and the method my professor used. Anyway, I'd be very interested to see what you guys come up with.