Have you pondered the Mean Value Theorem?
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.