Suppose we have $f(x,y)$ and $g(x,y)$ where $f$ is increasing in both $x$ and $y$ and $g$ is decreasing in both $x$ and $y$. Are there any simple conditions for $f$ and $g$ so that $h(x,y) = f(x,y)+g(x,y)$ to have only 1 maximum or minimum?
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A simple sufficient condition for $h(x,y)$ to have a unique maximum is that it is strictly concave and its value at some point is greater than its maximum on some closed curve surrounding that point. If $f$ is concave and $g$ is strictly concave (or vice versa), then $h$ is strictly concave.