Geometric interpretation of Generalized MVT

[rumjum]

Homework Statement

I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation

Homework Equations

[f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x)

The Attempt at a Solution

On substituting g(x) = x, we obtain the MVT which is nicely interpreted as a point in (a,b) where the tangent is parallel to the secant between the end points for a continuous differential function.

But, is there a nice interpretation for the generalized version?

 

[mighty2000]

I can provide a geometrical interpretation for the generalized MVT. First, let's review the meaning of the MVT. The Mean Value Theorem states that for a continuous and differentiable function f(x) on the closed interval [a,b], there exists a point c in (a,b) such that the slope of the tangent line at c is equal to the slope of the secant line connecting the endpoints a and b. This can be represented graphically as the tangent line and secant line being parallel to each other at point c.

Now, for the generalized MVT, the equation [f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x) can be interpreted as the ratio of the slopes of the tangent and secant lines at point c for two functions f(x) and g(x). In other words, the generalized MVT states that for two continuous and differentiable functions f(x) and g(x) on the closed interval [a,b], there exists a point c in (a,b) such that the ratio of the slopes of the tangent and secant lines for f(x) and g(x) at point c is equal to the ratio of their respective changes in value over the interval [a,b].

Visually, this can be represented as the tangent and secant lines for f(x) and g(x) intersecting at point c and forming a similar triangle with the corresponding changes in value for each function. The generalized MVT can also be seen as a generalization of the original MVT, where instead of just one function, we are considering the relationship between two functions.

In summary, the generalized MVT can be interpreted as a relationship between the slopes of tangent and secant lines for two functions at a given point, and the ratio of their respective changes in value over an interval. This interpretation can be useful in understanding the behavior of different functions and their relationships with each other.