If f(a)=g(a) and f(b)=g(b), prove parallel tangent lines exist

[NWeid1]

1. Homework Statement
(a) If f and g are differentiable functions on the interval [a,b] with f(a)=g(a) and f(b)=g(b), prove that at some point in the interval [a,b], f and g have parallel tangent lines.

(b) Prove that the result of part a holds if the assumptions f(a)=g(a) and f(b)=g(b) are relaxed to requiring f(b)-f(a)=g(b)-g(a).


2. Homework Equations



3. The Attempt at a Solution
I know to use the MVT, but besides that I'm lost.

 

[Mark44]

1. Homework Statement
(a) If f and g are differentiable functions on the interval [a,b] with f(a)=g(a) and f(b)=g(b), prove that at some point in the interval [a,b], f and g have parallel tangent lines.

(b) Prove that the result of part a holds if the assumptions f(a)=g(a) and f(b)=g(b) are relaxed to requiring f(b)-f(a)=g(b)-g(a).


2. Homework Equations



3. The Attempt at a Solution
I know to use the MVT, but besides that I'm lost.

Let h(x) = f(x) - g(x).
From the assumptions in part a, h(a) = h(b) = 0. Now you can use Rolle's Theorem, a special case of the MVT.

 

[NWeid1]

I already got it, but thanks, though.