Course:MATH102/Question Challenge/2008 December Q1.5
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Question
The function has local maXima (LX), local minima (LM) and inflection points(IP) as follows:
(a) LX: , LM: , IP: .
(b) LX: , LM: , IP: .
(c) LX: , LM: , IP: none
(d) LX: , LM: , IP: .
(e) LX: , LM: , IP: .
Hints
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Solutions
Solution |
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We take the first derivative to find the critical points:
From this we already see that whenever , and whenever . This already tells us that the positive critical point is a local maximum, and the negative critical point is a local minimum. (It is optional to plug in the actual values, as only the sign of actually matters, but if we do plug in, we would get the following:) At :
At
The function has a maximum at and a minimum at We ask where the second derivative changes sign (and hence where it also equals 0) to find the inflection points:
. We observe that indeed the second derivative changes sign at . Hence, the function has an inflection point at LX: x = 1, LM: x = -1, IP: x = 0 The correct answer is A) |