- #1
emjay66
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Help with this problem -- Proof with first and second derivatives
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]
N/A
[itex] [/itex]
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.
Homework Statement
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]
Homework Equations
N/A
[itex] [/itex]
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.