Thread: Tangent Lines to the Curves at Point C

This problem requires a graph to solve but I will try and explain what the graph looks like.

We have the interval [a,b] and the graph y= f(x) increasing on this interval but concave down and y=g(x) increasing concave up and both the graphs intersect each other on the points a,b.

Let both these functions be differentiable and c=the point where the curves are the farthest apart. What is special about the tangent lines to the curves at point c? Explain

*I have an idea that the tangent lines at point c where the curves are furthest apart have equal slope and possibly equal to the slope of the secant line between these 2 curves however I am unsure how to explain the mathematically if this happens to be the case.

Have you pondered the Mean Value Theorem?

thanks i assumed it had something to do with the mean value theorem but we just learned it so i dont fully understand it and how i would apply it to this problem

would it have something to do with the two curves sharing the same secant line then their would be at least one point that being 'c' in which the tangent line is parallel to the secant line