Yes, that looks like a good plan!
Show that the curvature is greatest at the endpoints of the major axis, and is least at the endpoints of the minor axis, for the ellipse given by
To solve this problem, do I need to compute ; and then find , to determine if the critical values coincide with the end points of the ellipse?
You should just solve for y as a function of x,
and then plug the derivatives y' and y'' into the equation for K. The definition of curvature should be a function of x right? Then differentiate it with respect to x.
Well how did you get here? Are you trying to differentiate it now that you have K? It looks real messy and it is difficult for me to sanity check because I don't know what derivatives you plugged in. the first and second derivatives I get for the function are and then
Ultimately you will have to plug those into the formula for K
Here's what it looks like when you plug them in:
It can be toyed with, I won't go through all of it but I will try to help with the first bit of the algebra, although I wonder if there is an easier way to accomplish this?
Now notice the denominator and numerator have the same power. So you can just multiply by the inverse of the denominator and cancel out a (4-x^2) factor:
Looks more manageable now huh?
Check and make sure you agree with everything though, but this looks like a function you can readily differentiate and look at critical numbers.