Prove that the equation x^7+x^5+x^3+1=0 as exactly one real solution. You should use Rolle's Theorem at some point in the proof.
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If we let , since and and is continuous, then by the intermediate value theorem, there's at least on root on . To prove there's only one root, suppose there are two, say . By Rolles theorem, there is a in where Further, must change sign crossing (i.e. goes up levels out and comes down or vice-versa). Now (it never changes sign). Thus, we have a contradiction - there can't be two roots.