Math 310-01/02

10-4-2017

1. Premises: .

Prove: B.

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.

However,

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.

*What would life be if we had no courage to attempt anything?*
- *Vincent Van Gogh*

Copyright 2017 by Bruce Ikenaga