Chapter 5.2, Problem 50E

To determine

To Prove: The fact that there is at least one point between a and b where the tangent to the graph of the f and g are parallel or the same line illustrate with the sketch.

Explanation of Solution

Given:

Assume that the $f$ and $g$ are differentiable on the $\left[a,b\right]$ and that $f\left(a\right)=f\left(b\right)$ and $f\left(b\right)=gb$ .

Calculation:

Consider that the mean value theorem show that the for continuous differentiable curve that connect the two points A and B there is the tangent line to the curve somewhere that is parallel to the connecting the two points that is chord AB. As bot the $f$ and $g$ passes through the points A and B then the tangent lines at the some point parallel to the chord AB and are parallel to each other.

Then, the graph obtained is shown in Figure 1

  

Figure 1