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