Necessary and sufficient conditions for the existence of a local minimum in a single- variable, unconstrained, nonlinear optimization problem are that the second derivative be negative at a point and at the same point the slope of the function be 0 .

Question

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The second derivative being negative at a point indicates that the function is concave down at that point, which is a necessary condition for a local minimum. The slope of the function being 0 at the same point indicates that the function is stationary at that point, which is also a necessary condition for a local minimum.

Necessary and Sufficient Conditions for the Existence of a Local