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Intervals where the function is increasing

  1. Jun 5, 2007 #1
    1. The problem statement, all variables and given/known data

    For the function f(x)=xln(x²), x≠0

    3. The attempt at a solution


    For the function to be increasing f'(x)>0


    Taking square roots of both sides


    How can I find the intervals from this?
    (Somebody has suggested to me that after taking the square root of 1/e the answer should be positive only rather than plus/minus, but I can't seem to find a reason why such would be the case)
  2. jcsd
  3. Jun 5, 2007 #2


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    Homework Helper

    note that if you accept that [tex]\log (x^2) = 2 \log (x)[/tex], then now RHS implies that x should be non-negative. That's the only possible reason I can think of that may dictate that you take the +ve root. But, it is clear that for your f(x) both +/- roots are good solutions and your interval is
    [tex](-\infty,-1/e) \cup (1/e, \infty)[/tex]
  4. Jun 5, 2007 #3
    yeah I got it!
    If I bring the two down to the front of the natural logarithm I will get

    With the modulus then I get |x|>(1/e) and hence the intervals (-∞, -1/e) and (1/e, ∞)

    Thanks! :tongue2:
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