1. Premises: .
2. Prove that the following curves do not intersect:
Suppose that the curves intersect in a point . Then satisfies both equations, and I may add the equations to get
Expand the left side and simplify:
Complete the square in x and y:
The left side is the sum of two squares, so it must be greater than or equal to 0. Consequently, it can't be equal to -4. This contradiction proves that the curves do not intersect.
3. Use Rolle's Theorem to prove that the function has at most two roots.
Suppose on the contrary that f has three roots: Say , where . By Rolle's theorem, there are numbers u and v, where
That is, f must have at least two critical points.
Thus, for only: f has only one critical point.
This contradiction shows that f can't have three roots. Therefore, f has at most two roots.
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