Slope of tangent line at a point

I posted this question in the Urgent Help section but nobody has been able to help me. If anyone could help I would really appreciate it.

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.

(Oh and point c is between a and b )