With this proof, I need to prove two things.
1) I need to prove the equation has solutions
2) Show the solution is unique
I know to prove the equation has solutions, I would like to use the intermediate value theorem, but I don't know exactly how to incorporate it. For number two, I figured how to prove the solution is unique , by some help. I used the Mean Value Theorem to prove it, but I would like to know how to prove it using the Rolle's Theorem. I would like to see it double sided, but I have never really went into depth on the Rolle's Theorem.