If f(a)=g(a) and f(b)-g(b), prove they will have parallel tan lines

[NWeid1]

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).

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

You know the rules. What have you tried?

 

[NWeid1]

I haven't, really. I know that by graphing specific examples I can intuitively confirm that it is true. But I'm struggling on how to rigorously prove it.

 

[LCKurtz]

Judging from your recent posts you have recently studied a theorem that has expressions like f(b)-f(a) in it haven't you?

 

[NWeid1]

Mvt?

 

[LCKurtz]

Yes. Here's my last hint before hitting the sack: Try applying that theorem to H(x) = f(x) - g(x).

 

[NWeid1]

Should I use f'(c)=(f(b)-f(a))/(b-a) and g'(c)=(g(b)-g(a))/(b-a)? Then how should i relate them to each other? Set them equal?

 

[HallsofIvy]

You've already said "f(a)=g(a) and f(b)=g(b)" so it is not necessary to "set them equal", they are equal!