Help with this problem - Proof with first and second derivatives

In summary, the problem involves showing that |f'(0)|+|f'(a)| \leq am, given the assumptions that |f''(x)| \leq m for each x in the interval [0,a] and that f has its largest value at an interior point of the interval. Using the Mean Value Theorem for integrals and the Fundamental Theorem of Calculus, it can be shown that |f'(a)| - |f'(0)| \leq m.a. However, the key to solving the problem lies in recognizing that there is an interior maximum at some point x=c, allowing for the use of f'(c) = 0.
  • #1
emjay66
10
0
Help with this problem -- Proof with first and second derivatives

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.
 
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  • #2
emjay66 said:

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.

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.
 
  • #3
Thanks. The correct interpretation of the FTC was a useful hint, as well as f'(c) = 0 for a c in [0,a].
 

1. What is the purpose of using first and second derivatives in proofs?

The first and second derivatives are used in proofs to analyze the behavior of a function. They provide information about the slope and curvature of the function, which can help in proving a statement or solving a problem.

2. How do I know when to use the first or second derivative in a proof?

The first derivative is used to find the slope of a function at a specific point, while the second derivative is used to find the concavity of a function at a specific point. Depending on the problem, you may need to use one or both derivatives in your proof.

3. Can I use the first and second derivatives to prove any statement about a function?

No, the first and second derivatives can only be used to prove statements that involve the slope or curvature of a function. They cannot be used to prove statements about the domain, range, or other properties of a function.

4. Are there any special rules or formulas for using first and second derivatives in proofs?

Yes, there are several rules and formulas that can be used when working with first and second derivatives. These include the power rule, product rule, quotient rule, and chain rule. It is important to familiarize yourself with these rules and practice using them in proofs.

5. Can I use computer software to help me with proofs that involve first and second derivatives?

Yes, there are many computer programs and calculators that can help you with calculations involving first and second derivatives. However, it is important to understand the concepts and theory behind these calculations in order to effectively use these tools in proofs.

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