Use Rolle's theorem to show that the function f(x) = 2X4 4X2 cannot take on the same value twice on the interval [1, 00). Assume there are two numbers a and b in [1, on) such that f(a) = f(b). By Rolle's Theorem, there exists at least one c E (a, b) C (1, on) such that f'(c) = 0, but f'(c) = 0 for this Function occurs when f'(X) = = 0 hence the only critical numbers are at X = 1, X = 0, and X = S . Since none of these numbers is in (1, 0c), there is no c E (a, b) such that f'(c) = 0, which is a contradiction. Thus fa) cannot equal f(b) for any a, b E [1, w), or in other words, the function cannot take on the same value twice in this interval. 4 Show that the function f(x) = has an absolute maximum but not an absolute minimum. x2+ 4 4 4 f ( x ) = ? v 0 for all x E IR. Since lim = , f does not take on an absolute minimum. x2 + 4:. ? X- 00 x2 + 4 2