Problem Statement: If where are real constants, prove that has at least one real root between and .
Proof: Consider the polynomial on . Then it follows that and which equals by our assumption. Furthermore, since is a polynomial with real coefficients it is differentiable on and so we may apply the Mean Value Theorem to on the interval . Applying MVT we see that there exists at least one point such that
So there is some such that . But we may compute since we know .
Thus, we have shown that there is at least one real root to on the interval .
Reflection: What a nice, sweet, and simple proof. I have to admit, when I first saw this problem I had no clue what to do. The trick came in remembering that I was in the chapter on differentiation and MVT, but that won’t happen on the qual…After completing the proof I understand a bit better how I would “see” the proof in the future. After realizing that the index of the numerator was plus the value of the denominator I started to see that differentiation could be helpful here. I think the hardest part of this proof is realizing that this is one of those times where you want a “helpful function”. After writing out and recognizing that the rest fell out.