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# Exam2_2 5 - < f(1< f 00(1 6 f(1< f 00(1< f(1...

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Taylor, Douglas – Exam 2 – Due: Oct 31 2007, 1:00 am – Inst: JEGilbert 5 By the Quotient rule, f 0 ( x ) = x 2 + 20 - 2 x ( x - 4) ( x 2 + 20) 2 = 20 + 8 x - x 2 ( x 2 + 20) 2 . The critical points of f occur when f 0 ( x ) = 0, i.e. , at the solutions of f 0 ( x ) = (2 + x )(10 - x ) ( x 2 + 20) 2 = 0 . Thus the critical points of f are x = - 2 and x = 10. To classify these critical points we use the First Derivative Test. But the sign of f 0 depends only on the numerator, so it is enough, therefore, to look only at a sign chart for (2 + x )(10 - x ): - 2 10 - - + From this it follows that f is decreasing on ( -∞ , - 2), increasing on ( - 2 , 10), and de- creasing on (10 , ). Consequently, f has a local maximum at x = 10 . keywords: local maximum, local minimum, critical point, quotient rule, First Derivative Test, rational function 011 (part 1 of 1) 10 points The graph of a twice-differentiable function f is shown in 1 Which one of the following sets of inequalities is satisfied by f and its derivatives at x = 1? 1. f 0 (1) < f 00 (1) < f (1) 2. f 00 (1) < f 0 (1) < f (1) 3. f 0 (1) < f (1) < f 00 (1) 4. f 00 (1) < f (1) < f 0 (1) correct
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Unformatted text preview: < f (1) < f 00 (1) 6. f (1) < f 00 (1) < f (1) Explanation: (i) Whether the point (1 , f (1)) lies above or below the x-axis determines the sign of f (1), and f (1) = 0 if the point lies on the x-axis. (ii) Whether the graph of f is increasing or decreasing at the point (1 , f (1)) deter-mines the sign of f (1), and f (1) = 0 when the tangent line to the graph is horizontal at (1 , f (1)). (iii) Whether the graph of f is concave up or concave down at the point (1 , f (1)) determines the sign of f 00 (1), and if the graph is changing concavity at (1 , f (1)), then f 00 (1) = 0. In the graph above, therefore, the inequali-ties f 00 (1) < f (1) < f (1) are satis²ed. keywords: concavity, decreasing, increasing, graph 012 (part 1 of 1) 10 points Find all intervals on which f ( x ) = x 2 ( x + 1) 3 is increasing....
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