Recently I was asked whether I could go over a visual proof of the Cauchy's Mean Value Theorem, as I had done for the Lagrange or simple version of the Mean Value Theorem (MFT). This was a very interesting question so I decided to go ahead and go over the graphical visualization of the theorem. In this video I show that the Cauchy or general mean value theorem can be graphically represented in the same way as for the simple MFT. The only difference is that the horizontal axis is not x, but a more general function of x, g(x). This difference causes the formulation to be more general, and the instantaneous slope to be formulated in the more general definition of the derivative, as shown in my last video. This is a very interesting topic to understand, but make sure to watch my earlier video on the general definition of derivative to get a better understanding of this!
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Cauchy’s Mean Value Theorem: Visual Proof
Recall the Cauchy's Mean Value Theorem:
Given the following conditions:
- f(x) and g(x) are continuous on the interval [a, b] and differentiable on (a, b).
- g'(x) ≠ 0 for all x in (a, b)
- g(a) ≠ g(b)
Then there is a number c in a < c < b, such that:
This is also known as the General Mean Value Theorem.
The Lagrange's Mean Value Theorem is a simple case of the Cauchy's MFT.
This is known simply as the Mean Value Theorem.
In my earlier videos I went over a visual proof of the MFT, but I did not go over a visual proof of Cauchy's MFT.
Recall that the simple MFT can be visualized as follows:
The Cauchy's MFT can be visualized in the same way BUT the horizontal axis is g(x).
In my last video, I went over a more general definition of derivative for when the horizontal axis is a general function of x, g(x), so make sure to watch that video!