Homework 06: Problem 8 Previous Problem Problem List Next Problem (1 point) +a+ .123 2:2 For what values of a and I) does ljm ( 3)0 sin(2m) b ) Z 0? b: Note: You can earn partial credit on this problem. Preview My Answers Submit Answers Previous Problem Problem List Next Problem (1 point) Use Rolle's Theorem and a proof by contradiction to show that the function f(m) = 6.:c9 6:6 14 does not have two real roots. Proof: Suppose f($) has two real roots a and 5 such that u.) : f(b) : Since the conditions of Rolle's theorem hold true for f on [(1, b], there exists at least one number c in the interval (a, b) such that f'(c) : However, the derivative f'(w) : is always ? v and, therefore, it is ? v for f'(a:) : This contradicts the conclusion of Rolle's Theorem and, therefore, f ? v have two real roots. Note: You can earn partial credit on this problem. Preview My Answers Submit Answers Homework 06: Problem 11 Previous Problem Problem List Next Problem (1 point) Consider the function at) : 22:3 33 on the interval [4, 4]. (a) The slope of the secant linejoining (4, f(4)) and (4, f(4)) is m = (b) Since the conditions of the Mean Value Theorem hold true, there exists at least one c on (4, 4) such that f'(c) : (c) Find 0. Note: If there is more than one answer, separate them with a comma. Homework 06: Problem 12 Previous Problem Problem List Next Problem (1 point) Find all critical numbers c of t) : 8t2/3 + t5/3. Note: If there is more than one critical number, separate them by a comma. Preview My Answers Submit Answers Homework 06: Problem 15 Previous Problem Problem List Next Problem 1 (1 point) Consider the function a: = . f( ) 5x2 +7 (a) f is concave up for a? E (b) f is concave down for a: E (c) The inection points of f occur at m = Note: Input U, infinity, and -innity for union, co, and 00, respectively. If there are multiple answers, separate them by commas. If there is no answer, input none. e: 8+e3' (1 point) Consider the function z] = (a) x) = (b) 3' is increasing for: E [c] f is decreasing for z E ((1) "he local minima of 1' occur at r. = (e) "he local maxima of 1' occur at :1: = m f" (I) = (g) f is concave up for m e (h) f is concave down for a E (i) The inection points of 1' occur at :I: : Note: Input U, innity, and -innity for union, 00, and oo, respectively. If there are multiple answers, separate them by commas. If there is no answer, input none