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I have two downward sloping concave functions of different slopes. They intersect at a point (let's call it x*).

I want to prove that if these functions are shifted downwards (upwards), then x* necessarily shifts left (right).

Any ideas on how to do this? Thanks!

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If both functions are shifted in the same direction by the same amount, then $x^*$ shifts in that same direction by that same amount. So you need some additional hypotheses. – Quinn Culver Jun 5 '12 at 14:10
In addition, (1) what do you mean by "different slopes" and (2) it's possible that two such functions might intersect at a point but that after translating one up, say, they might not intersect at all. As Quinn said, the question needs a bit of clarification. Welcome to SE, by the way. Hope we can help you. – Rick Decker Jun 6 '12 at 14:00

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