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Suppose $f(x,y)$ is a convex function and $$ f(x+\Delta x, y) < f(x,y), ~~~ f (x, y + \Delta y) < f(x,y)$$ Does this imply $$ f(x+\Delta x, y + \Delta y) < f(x,y)$$?

I am guessing the answer is no but I'm failing to come up with a counterexample.

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When you write $f(x+\Delta x, y) < f(x,y)$, do you mean that for all $(x,y)$ or just for some particular $(x,y)$? –  Robert Israel Feb 21 '13 at 2:15
    
Just for particular $x,y, \Delta x$. –  sree Feb 21 '13 at 2:16
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1 Answer

up vote 1 down vote accepted

Counterexample: $f(x,y) = (x+y)(x+y-3)$, $(x,y) = (0,0)$, $\Delta x = 2$,$\Delta y = 2$

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Thank you (for some reason, I cannot upvote your answer). Can you provide some intuition? How did you come up with this answer? –  sree Feb 21 '13 at 2:21
    
It's really a one-dimensional situation, but you wanted two-dimensional. Convex functions can go down and then up. –  Robert Israel Feb 21 '13 at 3:52
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