Does the property of non-increasing slope be generalized to a concave function for multiple variables?

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cenjene9gw · Accepted Answer

A common characterization requires you take two derivatives and work with the Hessian. I&#039;m going to phrase this in the convex case (but the concave case is analogous, just multiply everything by   −  1).By calculus, &quot;      f    ′   is always increasing&quot; is the same as saying &quot;      f    ″   is always positive&quot; (provided       f    ″   exists). You have observed that this property holds for convex functions. More generally, the following is a common result:If a real-valued function defined on a real Hilbert space (e.g.             R        N  ) has a well-defined Hessian,   H then  f      is convex      ⇔      H      is positive semidefinite  .

Does the property of non-increasing slope be generalized to a concave function for multiple variables?

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