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How Derivatives Affect the Graph (Proving Question)

  1. Nov 5, 2009 #1
    1. The problem statement, all variables and given/known data

    (a) If f and g are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I.

    (b)Show that part (a) remains true if f and g are both decreasing.

    2. Relevant equations

    -

    3. The attempt at a solution

    (a)
    f>0, f'>0, f''>0
    g>0, g'>0, g''>0

    (fg)'=f'g+fg'>0
    (fg)''=f''g+2f'g'+fg''>0 and so fg is concave upward on I.

    (b)
    f'<0, f''<0
    g'<0, g''<0

    (fg)''=f''g+2f'g'+fg''

    I have 2f'g'>0 but f''g <0 and fg''<0. Then I'm stuck here. The answer mentioned that (fg)"> or equals to f''g+fg''>0 but I ain't sure how that came about.

    Thanks!
     
  2. jcsd
  3. Nov 5, 2009 #2
    They are still positive and concave, aren't they?
     
  4. Nov 5, 2009 #3
    If f and g are both decreasing, then f'<0 and g'<0. Is that right? And if both the slopes are decreasing, then f''<0 and g''<0?
     
  5. Nov 5, 2009 #4
    That does not mean that the slopes are decreasing. Suppose for example f = 1/x.
     
  6. Nov 5, 2009 #5
    It says in the text that if f'>0, then f is increasing and if f'<0, then f is decreasing on the interval?
     
  7. Nov 5, 2009 #6
    ... and the text would be correct. However, f is not the slope, f is the function. f' is the slope and knowing that the slope is negative does definitely not tell you how it is changing.
     
  8. Nov 5, 2009 #7
    Oh okay. Thanks.
     
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