Science:Math Exam Resources/Courses/MATH100/December 2010/Question 04 (d)
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Question 04 (d)
Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.
NOTE: Another notation for tan-1(x) is arctan(x)
Determine open intervals where the graph of ƒ is concave upwards or concave downwards, and the x-coordinates of all inflection points (if any).
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How does concavity relate to the derivatives of the function f(x)?
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To determine the intervals of concavity, we must evaluate ƒ''(x) and determine where it is positive and negative:
Evaluating the second derivative of f gives
From this we see that the possible inflection points are x = 0. We need only confirm that the sign of the concavity changes across x = 0. We must also consider concavity changes across x = 1 since ƒ is not differentiable there.
When x < 1, the second derivative is always negative since -12x2 is always negative.
When x > 1, the second derivative is always negative as well since in that case, -2x is always negative.
From these results we can see that ƒ(x) is concave down for all x except at the critical points. In other words, ƒ(x) is concave down on and so, there are no inflection points.