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I have a question regarding the Arrow-Enthoven-Theorem which is used to find sufficient conditions for a global maximum of a quasi-concave function under quasi-convex restrictions also see: http://michaelcarteronline.com/FOME/pdf/ConstraintQualification.pdf, P. 4). In "Fundamental Methods of Mathematical Economics" from Chiang and Wainwright page 426 it is stated that for one condition: "(d) any one of the following is satisfied:

(d-i) $f_j (x^*) <0 $ for at least one variable $x_j$

(d-ii) $f_j (x^*)>0$ for some variable $x_j$ that can take on a positive value without violating the constraints

(d-iii) the $n$ derivatives $f_j(x^*)$ are not all zero, and the function $f(x)$ is twice differentiable in the neighbordoof of $x^*$ (i. e. all the second-order partial derivatives of $f(x)$ exist at $x^*$)"

In this case $f(x_j) $ is the objective function with $x_j =1, 2, 3, ..., n$ and $f_j$ is the partial derivative of $\partial f/ \partial x_j $. The value $x^*$ is the optimal value which maximizes $f(x)$.

My problem is the following: Which conditions of the three above can I use if I only have a function with one variable, namely $x_1$? The first condition can't be satisfied because the derivative of $f$ in $x^*$ should be zero and the same goes for the second condition. In the third condition I don't have any other derivatives for $f$ because I only have one $x$ value. Although the function is twice differentiable in the neighborhood of $x^*$. Maybe I can only use this last statement to meet the necessary conditions for the Arrow-Enthoven Theorem?

Thank you for your comments.

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