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Math Help - showing a multivariate function is unimodal

  1. #1
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    showing a multivariate function is unimodal

    I am trying prove that a function has a unique optimal solution. It is a function with two variables, and it is not necessarily well-behaved, in particular is not concave. Can I argue the following?
    The vector of first order derivatives is decreasing in the vector of variables, hence the function is unimodal and would have a unique optimal solution.
    In other words, for univariate functions, we can show the unimodality by argueing that the first derivative is decreasing. Can I generalize it to the multivariate case as described above?

    Thanks for any help provided.
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  2. #2
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    What do you mean by a vector is "decreasing"? There is no linear ordering of vectors.
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  3. #3
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    I am sorry, let me rephrase my question: Using only the first derivatives and the cross partial derivatives, how can I show that a function of two variables is unimodal? More specifically, if the function is z = z(x,y), I am talking about partial derivative with respect to x, z_x, partial derivative with respect to y, z_y, and the cross partial derivative z_xy.
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