# 12Find the absolute maximum value on (0, co) for f(x) = 18-12x-XSelect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.at x-O A. The absolute maximum isO B. There is no absolute maximum.lirmIravClick to select and enter your answer(s) and then click Check Answer.rarAll parts showingClear Allhere to search

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Asked Nov 11, 2019
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## Expert Answer

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Step 1

f(x) = 18 - 12x - 12/x

f'(x) = -12 + 12 / x2

Step 2

The critical points are given by f'(x) = -12 + 12 / x2 = 0

Hence,  12 / x2 = 12

Hence, x2 = 1

Hence, x = 1 or -1

However the given interval for x is (0, ∞)

x = -1 falls outside this interval, hence  x = 1 is the only critical point in the given interval.

Step 3
Hence the critical point is x = 1 and the end points of the interval are x = 0 and

f(0) = 18 - 12 x 0 - 12 / 0 = -∞

f(1) = 18 - 12 x 1 - 12 / 1 = -6

f(∞) = 18 - 12 x ∞ - 12 / ∞ = -∞

Since, f(1) is the highest value, hence, the absolute maxima occurs at x = 1 and the maxima is -6

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