Solve Mean Value Theorem Problem on [1,4]

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Homework Help Overview

The discussion revolves around applying the Mean Value Theorem to the function f(x) = x(x^2 - 8) - 5 over the interval [1, 4]. Participants are tasked with finding a number C in the interval (1, 4) that satisfies the theorem's conditions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the steps involved in applying the Mean Value Theorem, including expanding the function, finding its derivative, and calculating the average rate of change. Questions arise about the correct interpretation of the theorem and the calculations involved, particularly regarding the equality to 1 and the values of f(1) and f(4).

Discussion Status

There is an ongoing exploration of the correct approach to find C, with some participants clarifying misunderstandings about the theorem's application. Guidance has been offered regarding the need to equate the derivative to the average rate of change rather than to 1.

Contextual Notes

Some participants express confusion about the steps and calculations, indicating a lack of clarity on how to proceed with the problem. The original poster has provided some calculations but is unsure about the next steps and the relevance of their findings.

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Homework Statement


Given the function f(x)= x(x^2-8)-5 satisfies the hypothesis of the Mean Value Thereom on the interval [1,4], find a number C in the interval (1,4) which satisfies this thereom.




Homework Equations



f'(c) = f(b)-f(a) / b-a

The Attempt at a Solution



1) Expand the equation first
2) Find the first derivative.
3) Equal the equation to 1

Apparently, I got the wrong answer. What am I doing wrong?
PLEASE HELP.
 
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That's a great strategy. Impossible to tell you how you got the wrong answer until you tell us what you got for (f(b)-f(a))/(b-a) and C.
 
What do you mean by "equal the equation to 1"? Don't you have to find both f(1) and f(4) then use the Mean Value theorem to find c?
 
I thought the OP meant "equal the equation" to (f(b)-f(a))/(b-a). I may have been extrapolating on that.
 
Hm.. I just realized it and I'm stuck. I don't know what to do or what I'm trying to get.. HAHA

On the bright side, I do have the C value and the value for (f(b)-f(a))/(a-b):

(f(b)-f(a))/(a-b)
( 27 + 12 )/(4-1) = 13

C value = x^3-8x-5
f' = 3x^2-8
3x^2-8 = 1
3x^2 = 9
9 / 3 ^1/2
= 3^1/2

So, what to do next? Or what the heck am I suppose to get?
 
I found out the tangent line is at (3^1/2, -13.66) which is parallel to the secant line through (1, -12) and (4, 27)

Now, I don't even know if that helps.. but there it is. Lol!
 
Defennnder said:
What do you mean by "equal the equation to 1"? Don't you have to find both f(1) and f(4) then use the Mean Value theorem to find c?

What I meant by equal the equation to one was that getting the derivative of the equation and equalling it to 1.

1 = 3x^2 -8
 
Don't "equal it to 1". Equal it to (f(b)-f(a))/(b-a)=13. Read the mean value theorem again.
 
HAHA. Thanks. That's all I needed to know. You've solved one of my many problems, AGAIN! THANKS!
 
  • #10
Gotta admit, you resolve your own problems quickly. Hope this is a short lived phase of confusion.
 

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