Solve Mean Value Theorem Problem on [1,4]

In summary, the homework statement given is that the function f(x)=x(x^2-8)-5 satisfies the hypothesis of the Mean Value Thereom on the interval [1,4]. A number C in the interval (1,4) which satisfies this theorem is given. However, the OP got the wrong answer and does not know how to solve for c.
  • #1
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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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  • #2
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.
 
  • #3
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?
 
  • #4
I thought the OP meant "equal the equation" to (f(b)-f(a))/(b-a). I may have been extrapolating on that.
 
  • #5
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?
 
  • #6
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!
 
  • #7
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
 
  • #8
Don't "equal it to 1". Equal it to (f(b)-f(a))/(b-a)=13. Read the mean value theorem again.
 
  • #9
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.
 

What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) where the slope of the function is equal to the average slope of the secant line between points (a, f(a)) and (b, f(b)).

What is the purpose of solving Mean Value Theorem problems?

Solving Mean Value Theorem problems allows us to find precise values for the average rate of change of a function over a given interval. This can be useful in many real-world applications, such as finding the average speed of a moving object or the average growth rate of a population.

What are the steps for solving a Mean Value Theorem problem on a closed interval [a, b]?

The steps for solving a Mean Value Theorem problem on a closed interval [a, b] are as follows:
1. Verify that the function is continuous on [a, b] and differentiable on (a, b).
2. Find the average rate of change of the function over [a, b] using the formula (f(b) - f(a)) / (b - a).
3. Find the derivative of the function using the rules of differentiation.
4. Set the derivative equal to the average rate of change and solve for the unknown variable.
5. The value of the unknown variable found in step 4 is the point c where the Mean Value Theorem is satisfied.

Can the Mean Value Theorem be used to find the average value of a function?

No, the Mean Value Theorem only allows us to find a specific point on the function where the average rate of change is equal to the average slope of the secant line. To find the average value of a function over an interval, we need to use the Mean Value Theorem for Integrals.

What are the common mistakes to avoid when solving Mean Value Theorem problems?

Some common mistakes to avoid when solving Mean Value Theorem problems are:
- Forgetting to verify the conditions of continuity and differentiability on the given interval.
- Using the wrong formula for the average rate of change or derivative of the function.
- Not simplifying the equation and solving for the unknown variable.
- Confusing the Mean Value Theorem with the Mean Value Theorem for Integrals.
It is important to carefully follow the steps and double-check the calculations to avoid these mistakes.

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