If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ambiguity by comparing the signs of the first derivatives in intervals bordering the point?

For example, given an equation of the form $z = f(x, y)$, can you slice its 3D graph along the $x$- and $y$-axes, and within each slice, check whether the function is increasing or decreasing in the intervals above and below the point, then somehow combine the information into a complete picture?

If so, is this technique ever necessary? For example, it appears to me that the matrix from quadratic form minimization (circled below) is actually always the Hessian matrix of the quadratic expression.

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However, this class of Hessian matrix always has purely constant entries, so if (counterfactually) it was positive or negative semidefinite, the ambiguity could not be resolved by taking third or higher partial derivatives of the function. Could the interval analysis under consideration be necessary in a case where this sort of condition arises?


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