Help with graphing calculus problem?

Question

Sketch on the xy-plane the graphs of any two functions f and g that on some interval [a, b], and intersect each other at a and b: that is f(a) = g(a) and f(b)=g(b)

Does there have to be some point c in (a, b) where the slope of the tangent line to the equals the slope of the tangent line to the graph of g? Explain how you know, and if such a point exists exists, mark it on your graph.

Answer 1

If we assume that f(x) and g(x) are differentiable on [a,b]. then h(x) = f(x) - g(x) is also differentiable on [a,b].

Since f(a) = g(a) and f(b) = g(b), h(a) = h(b) = 0

So, by Rolle's Theorem, there is at least one number c in the interval (a, b), such that h'(c) = f'(c) - g'(c) = 0.

That is, f'(c) = g'(c).

Answer 2

Since you do not specify that the functions must be differentiable on the interval then the answer is "no" there does not need to be a point "c" where the two tangent lines are the same slope. Example:

Notice that the slope of f is zero everywhere and the slope of g is either positive or negative, never zero. BUT, g is not differentiable everywhere on (a, b)