Q7Mathematics I

Question

State the necessary and sufficient conditions for the minimum of a function $f (x, y)$ .

Answer 1

In multivariable calculus, determining the absolute maximum or minimum of a continuous function $f (x, y)$ involves checking both necessary and sufficient mathematical conditions using partial derivatives.

1. Strict Necessary Conditions (Stationary Points): For any point $(a, b)$ to even be considered a candidate for a maximum or minimum, the first-order partial derivatives evaluated at that exact point must both be completely equal to zero. These points are technically termed stationary or critical points.

\frac{\partial f}{\partial x} = 0 and \frac{\partial f}{\partial y} = 0

2. Strict Sufficient Conditions (for a Local Minimum): To definitively confirm that the identified stationary point is a local minimum, we must systematically evaluate the second-order partial derivatives. We define standard notational variables as $r = f_{xx}$ (second derivative with respect to x), $s = f_{x y}$ (mixed partial derivative), and $t = f_{y y}$ (second derivative with respect to y).

The strict mathematical conditions that unequivocally guarantee the stationary point is a local minimum are:

r t - s^{2} > 0 and r > 0

(Note: If $r t - s^{2} > 0$ but $r < 0$ , the point would strictly be a local maximum. If $r t - s^{2} < 0$ , it is definitively a saddle point.)

Q7Mathematics I

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