Can the MVDT be used to prove an inequality in economics?

[Bipolarity]

I'm trying to prove something small in economics using MVDT but can't seem to work it out:

So consider: S(x) and D(x) such that:

S' > 0 > D'
S(x_{1}) = D(x_{1}) = Q_{1}
S(x_{2}) + \epsilon = D(x_{2}) + \epsilon = Q_{2}

Then can I prove the following?
x_{2} = x_{1}
Q_{2} > Q_{1}

I've tried to define a function that is the difference between S and D and then apply the mean value theorem, but it seems like I can't get where I want.

By the way there may not be an answer since I made the problem up. But instinct tells me this is doable, I just can't figure out how.

Thanks for your help.

BiP

 

[scurty]

Intuitively I would think because S(x) is strictly increasing and D(x) is strictly decreasing, that if they intersect at one point (Q_1) then they cannot intersect at another point. So if you assume your assumptions, I would say, yes, x_1 = x_2 and because of that, Q_2 > Q_1, namely, Q_2 = Q_1 + \epsilon.

Here would be my pseudo proof: Show that x_1 = x_2 by contradiction (by assuming intersection at two points). Then show Q_2 = Q_1 + \epsilon which implies Q_2 > Q_1.

Someone else can chime in with a better proof idea if they want.

 

[Bipolarity]

Yes my intuition was also similar, I just wanted to formalize it. Thanks though!

BiP