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Help with this problem - Proof with first and second derivatives

  1. May 6, 2014 #1
    Help with this problem -- Proof with first and second derivatives

    1. The problem statement, all variables and given/known data
    I'm stuck on this problem and I'm not sure what I'm missing. The problem states:
    Assume that [itex] |f''(x)| \leq m[/itex] for each [itex]x [/itex] in the interval [itex][0,a] [/itex], and assume that [itex] f[/itex] takes on its largest value at an interior point of this interval. Show that [itex]|f'(0)|+|f'(a)| \leq am [/itex]. You may assume that [itex]f'' [/itex] is continuous on [itex] [0,a][/itex]

    2. Relevant equations
    N/A
    [itex] [/itex]

    3. The attempt at a solution
    I first observed that using the Mean Value Theorem for integrals and letting [itex]c[/itex] be a number in the interval [itex][0,a] [/itex], I can obtain
    [itex]
    \int_0^a|f''(t)|\,dt = |f''(c)|.(a-0) \leq m.a
    [/itex]
    I also observed that, using the Fundamental Theorem of Calculus, I can also obtain
    [itex]
    \int_0^a|f''(t)|\,dt = |f'(a)| - |f'(0)|
    [/itex]
    which would imply
    [itex]
    |f'(a)| - |f'(0)| \leq m.a
    [/itex]
    I know that
    [itex]
    0\leq|f'(a)| - |f'(0)| \leq |f'(a) - f'(0)| \leq |f'(a)| + |f'(0)|
    [/itex]
    but I haven't been able to determine what the next step is. Based on the above information, I can't see how I can deduce the answer from what I have so far, so I'm clearly missing something. Any Hints would be very welcome.
     
  2. jcsd
  3. May 6, 2014 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The fundamental theorem of calculus does NOT tell you that ##\int_0^a|f''(t)|\,dt = |f'(a)| - |f'(0)|##. It tells you that##\int_0^c f''(t)\,dt = f'(c) - f'(0)##. There is an interior maximum at some point x=c. You haven't used that yet. Use that.
     
  4. May 6, 2014 #3
    Thanks. The correct interpretation of the FTC was a useful hint, as well as f'(c) = 0 for a c in [0,a].
     
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