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Math(Please help)

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What is the domain? Use a graphing utility to determine the intervals in which the function is increasing and decreasing and approximate any relative maximum or minimun values of the function.

g(x) = 12 Ln x / x

My answer was: Domain = (12,0)
decreasing = (3,-2) and increasing = (-2,3) but these were wrong.

  • Math(Please help) - ,

    First of all we can only take the ln of positive numbers,

    so x > 0

    (notice that also takes care of the possibility of dividing by zero)

    g'(x) using the quotient rule was
    (12 - 12lnx)/x^2

    let's set that equal to zero for max/mins
    12lnx = 12
    lnx = 1
    x = e

    So when x = e the curve is neither increasing nor decreasing.
    We could take the second derivative , but an easier way is to simply test a value less than and a value greater than e, and see if it is positive or negative.
    We don't need that actual value

    let x = 2
    g'(2) = (12 - 12ln2)/4 , clearly positive , since ln2 is negative)
    let x = 3
    g'(3) = (12-l2ln3)/9 , which is negative

    also if x = 3
    g(e) = 12lne/e = 12/e = appr. 12/3 = appr.4
    (calculator value =4.4145)

    so...

    domain : x > 0
    increasing : 0 < x < e
    decreasing: x > e

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