Strictly increasing concave function
= Strictly increasing concave function
Let a differentiable <concave function> on all of $\mathbb R^d$ be strictly increasing in each coordinate separately. Every coordinate derivative is strictly positive. Indeed it is nonnegative by monotonicity and nonincreasing along its coordinate by <concavity>; a zero value would force zero derivative on the remaining positive ray and hence a constant segment. The global-domain assumption matters at endpoints of restricted domains.