A question about mean value theorem

In summary, the conversation discusses the problem of finding the value of c using the mean value theorem for a given function. The function is defined as f(x) = 2x^3 - x + 1 for x ∈ [0,1], and f(x) = 3x^2 - x for x ∈ (1,3]. The conversation mentions that the value of c can be found when the slope of the tangent is equal to the slope of the chord. The solution is given as c = 13/9, and there are other possible solutions of c = 2 or c = 1/(sqrt 3). The person solving the problem used different methods to find these solutions. They also mention that it is
  • #1
m.medhat
37
0

Homework Statement


hello ,
if f(x) is a function which satisfies the mean value theorem , where :-
[tex]f(x) = \left\{ {\begin{array}{*{20}c}
{2x^3 - x + 1\quad \quad x \in [0,1]} \\
{3x^2 - x\quad \quad \quad x \in (1,3]} \\
\end{array}} \right.[/tex]

find the value of (c) by using the mean value theorem , where (c) is the points in which the slope of tangent is equal to the slope of chord .
I want the steps of the solution please .



Homework Equations





The Attempt at a Solution


i solve with some method and find that c=13/9
and I solve with other method and find that c=2 or c=1/(sqrt 3)



very thanks >>
 
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  • #2
What method did you try to solve it with? How did you come up with those two solutions? Please show us what you've done...
 
  • #3
It is not forbidden that in more than 1 point they have the same slope.
 

1. What is the mean value theorem?

The mean value theorem is a fundamental theorem in calculus that states that for a continuous and differentiable function on an interval, there exists at least one point in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change over the interval.

2. What is the significance of the mean value theorem?

The mean value theorem provides a way to relate the instantaneous rate of change of a function to its average rate of change over an interval. It is often used in proving other important theorems in calculus and is also used in real-world applications such as optimization problems.

3. How is the mean value theorem applied in calculus?

The mean value theorem is often used to prove other theorems in calculus, such as the fundamental theorem of calculus, Rolle's theorem, and the first and second derivative tests. It is also used in finding the maximum and minimum values of a function and solving optimization problems.

4. Can the mean value theorem be used for all functions?

No, the mean value theorem can only be applied to continuous and differentiable functions. This means that the function must be continuous on the interval and have a derivative at every point in the interval.

5. Where did the mean value theorem originate?

The mean value theorem was first stated and proved by French mathematician Augustin-Louis Cauchy in the early 19th century. However, it was previously discovered and used by mathematicians such as Rolle and Fermat in the 17th century.

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