Mean value theorem(mvt) to prove the inequality

In summary, the conversation discusses using the mean value theorem to prove the inequality |sin a - sin b| ≤ |a - b| for all a and b. The speaker is unfamiliar with using the theorem for inequalities and is looking for hints. The expert suggests using the mean value theorem and the fact that |cos(x)| ≤ 1 to solve the problem.
  • #1
Khayyam89
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Homework Statement


Essentially, the question asks to use the mean value theorem(mvt) to prove the inequality: [tex]\left|[/tex]sin a -sin b [tex]\right|[/tex] [tex]\leq[/tex] [tex]\left|[/tex] a - b[tex]\right|[/tex] for all a and b


The Attempt at a Solution



I do not have a graphing calculator nor can I use one for this problem, so I need to prove that the inequality basically by proof. What I did was to look at the mvt hypotheses: if the function is continuous and differetiable on closed and open on interval a,b, respectively. However, the problem I am having is that I am getting thrown off by the absolute values and the fact that I've never used mvt on inequalities. I know the absolute value of the sin will look like a sequence of upside-down cups with vertical tangents between them. Hints most appreciated.
 
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  • #2
The mean value theorem states that in an interval [a,b]:

[tex]f'(c) = \frac{ \sin b - \sin a}{b-a} = \cos(c) [/tex]

Now put absolute value signs there and make use of [tex]|\cos(x)| \leq 1[/tex]
 

1. What is the Mean Value Theorem (MVT)?

The Mean Value Theorem states that for a continuous function on an interval, there exists a point within the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.

2. How is the MVT used to prove inequalities?

The MVT can be used to prove inequalities by showing that the slope of the tangent line is always less than or greater than the slope of the secant line for a given interval. This can be done by setting up equations and using algebraic manipulations.

3. What are the requirements for using the MVT to prove an inequality?

The function must be continuous on the given interval and differentiable on the open interval. Additionally, the endpoints of the interval must be distinct and the function must satisfy the necessary conditions for the MVT to hold.

4. Can the MVT be used to prove all types of inequalities?

No, the MVT can only be used to prove inequalities that involve continuous functions on an interval. It cannot be used for discrete functions or inequalities involving discontinuous functions.

5. Are there any limitations to using the MVT to prove inequalities?

Yes, the MVT can only prove inequalities for one variable. It cannot be used to prove inequalities for multivariable functions. Also, the MVT can only prove strict inequalities (less than or greater than), not non-strict inequalities (less than or equal to or greater than or equal to).

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