Applying the Mean Value Theorem

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Homework Help Overview

The discussion revolves around applying the Mean Value Theorem (MVT) to analyze the behavior of a function f(x) based on its derivative graph. Participants are exploring the implications of the MVT in relation to specific values of the function within a defined interval.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to understand how to apply the MVT to derive inequalities and behavior of the function f(x) based on its derivative. Some participants question how to interpret the graph of f'(x) to determine the behavior of f(x), while others discuss the significance of critical points and the implications of decreasing and increasing intervals.

Discussion Status

Participants are actively engaging with the problem, raising questions about the application of the MVT and the interpretation of the derivative graph. There is a focus on understanding the behavior of f(x) in relation to its derivative, and some guidance has been offered regarding the first derivative test and critical points.

Contextual Notes

The original poster has specific values and intervals for f(x) and its derivative, but there may be uncertainties regarding how to apply the MVT effectively to answer the posed questions. The discussion is framed within the constraints of the problem, which requires careful reasoning based on the provided graph.

soe236
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Hey,
I know the basic definition of the MVT, but I'm having a lot of trouble applying it to this problem. I would greatly appreciate any kind of help or guidance.

A graph of the derivative of f(x) is displayed below. Information about the function f(x) is known only for -2.5 < x < 3.5. Also f(-2) = 1. Consider the graph carefully, and consider the information in both the numbers and the shapes of the graph (both "quantitative" and "qualitative" information).

the graph:
http://img148.imageshack.us/img148/5224/graphfx6.jpg

a) Explain why -2 < f(0) < -1. Look carefully at the graph and make estimates using the MVT. Explain the steps of your reasoning in detail.
b) Explain why f(3) > 4 +f(1). Again, use the MVT and explain your reasoning in detail.
c) How big and how small can f(1) - f(0) be?
d) Use the information in a), b), and c) to explain why f(3) must be positive.
e) Explain why f(x) = 0 must have a solution between 0 and 3. Use the IVT (intermediate value th.) and the
information obtained in previous parts of this problem.
 
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Starting from x = -2, how does f(x) behave, can you tell by looking at f'(x)?
 
EnumaElish said:
Starting from x = -2, how does f(x) behave, can you tell by looking at f'(x)?

well from x=-2 to about x=.5, f'(x) is negative which means f(x) is decreasing and f(x) is increasing on approx [0.5,3.5] ..do i do something with the first derivative test? I just don't know what the question is asking for or atleast what I have to find to answer it
 
and the critical point is at 0.5 if that's of any use?
 

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