Cauchy Mean Value Theorem/Geometrical Interpretation

Geometrical Interpretation of Cauchy Mean Value Theorem
Consider two functions $f \left({x}\right)$ and $g \left({x}\right)$ continuous on the interval $\left[{a \,.\,.\, b}\right]$ and differentiable on $\left({a \,.\,.\, b}\right)$.

For every $x \in \left[{a \,.\,.\, b}\right]$, we consider the point $\left({f \left({x}\right), g \left({x}\right)}\right)$.

If we trace out the points $\left({f \left({x}\right), g \left({x}\right)}\right)$ over every $x \in \left[{a \,.\,.\, b}\right]$, we get a curve in two dimensions, as shown in the graph:


 * [[File:Cauchy's Mean Value Theorem.png]]

In the drawing, the slope of the red line is $\dfrac{g \left({b}\right) - g \left({a}\right)} {f \left({b}\right) - f \left({a}\right)}$.

This is because:
 * $\dfrac {\Delta y} {\Delta x} = \dfrac {g \left({b}\right) - g \left({a}\right)} {f \left({b}\right) - f \left({a}\right)}$

assuming that the vertical axis, which contains the value of $g \left({x}\right)$, is the $y$-axis.

The slope of the green line is $\dfrac {g' \left({c}\right)} {f' \left({c}\right)}$.

This is because:
 * $\left.{\dfrac {\mathrm d g}{\mathrm d f} }\right\vert_{x \mathop = c} = \left.{\dfrac{\mathrm d g / \mathrm d x}{\mathrm d f / \mathrm d x} }\right\vert_{x \mathop = c} = \dfrac{g' \left({c}\right)} {f' \left({c}\right)}$

The drawing illustrates that for the value of $c$ chosen in the pictures, the slopes of the red line and green line are the same.

That is:
 * $\dfrac {g \left({b}\right) - g \left({a}\right)} {f \left({b}\right) - f \left({a}\right)} = \dfrac {g' \left({c}\right)} {f' \left({c}\right)}$