6. [-/6 Points] DETAILS MY NOTES Use Rolle's theorem to show that the function f(x) = 3x* - 6x2 cannot take on the same value twice on the interval [1, co). Assume there are two numbers a and b in [1, co) such that f(a) = f(b). By Rolle's Theorem, there exists at least one c E (a, b) C (1, co) 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 = Since none of these numbers is in (1, co), there is no c E (a, b) such that f'(c) = 0, which is a contradiction. Thus f(a) cannot equal f(b) for any a, b E [1, co), or in other words, the function cannot take on the same value twice in this interval. 7. [-/6 Points] DETAILS MY NOTES 5 Show that the function f(x) = has an absolute maximum but not an absolute minimum. x 2 + 5 5 f(x ) = x2 + 3 0 for all x E IR. Since lim X -+ 00 x2+ 5 , f does not take on an absolute minimum. 5 Since x + 5 t follows that f(x) * 2 + 5 7 0 1. f(0) = and that is the absolute maximum. Submit Answer 8. [-/5 Points] DETAILS MY NOTES Let f(x) = x'. Verify the mean value theorem by finding a ce (-2, 1) such that F'(c) = (1) - f(-2) 1 - (-2)