An even degree polynomial function, denoted as f(x), has either an absolute maximum or minimum due to its behavior as x approaches positive or negative infinity. The proof relies on the Extreme Value Theorem (EVT) and the Bolzano-Weierstrass theorem, establishing that f(x) is continuous and bounded on a closed interval [m, M]. The discussion highlights that the limits of f(x) as x approaches infinity are the same, confirming the existence of extrema. The leading coefficient's positivity is also crucial in determining the polynomial's behavior.
PREREQUISITESMathematicians, students studying calculus and real analysis, and anyone interested in understanding the properties of polynomial functions and their extrema.
Dick said:I just meant that the guts of the proof of the EVT is BW added to the definition of continuity. That's all. Not that they are somehow formally equivalent. And yes, your proof was on the right track. I was thinking EVT was the continuous function bounded theorem. Just confused.