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c1-t4t-a - NAME Brief Answers TEST-4T/MAC2311 Page 1 of 5...

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NAME: Brief Answers TEST-4T/MAC2311 Page 1 of 5 Read Me First: Show all essential work very neatly. Use correct notation when presenting your computations and arguments. Write using complete sentences. Be careful. Remember this: "=" denotes "equals" , " " denotes "implies" , and " " denotes "is equivalent to". Do not "box" your answers. Communicate. Show me all the magic on the page. 1. (16 pts.) Fill in the blanks of the following analysis with the correct terminology. Let f(x) = x 4 - 8x 3 . Then f (x) = 4x 3 - 24x 2 = 4(x - 0) 2 (x - 6). Consequently, x = 0 and x = 6 are critical (stationary) points of f. Since f (x) > 0 for 6 < x, f is increasing on the set (6, ). Also, because f (x) < 0 when 0 < x < 6 or x < 0, and f is continuous, f is decreasing on the interval (- , 6). Using the first derivative test, it follows that f has a(n) relative minimum at x = 6, and neither a relative max nor a relative min at x = 0. Since f (x) = 12x 2 - 48x = 12x (x - 4), we have f (0) = 0, f (4) = 0, f (x) < 0 when 0 < x < 4, and f (x) > 0 when x > 4 or x < 0. Thus, f is concave down on the interval (0,4), f is concave up on the set (- ,0) (4, ), and f has inflection points at x = 0 and x = 4. 2. (4 pts.) Rolle’s Theorem states that if f(x) is continuous on [a,b] with f(a) = f(b) = 0 and differentiable on (a,b), then there is a number c in (a,b) such that f (c) = 0. Give an example of a function f(x) defined on [-1,1] with f differentiable on (-1,1) and f(-1) = f(1) = 0 but such that there is no number c in (-1,1) with f (c) = 0. [Hint: Which hypothesis above must you violate??] // A suitable example clearly must fail to be continuous on the interval [-1,1] and yet satisfy the remaining hypotheses. Here is one such, defined in pieces, of course.
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