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Handbook of mathematics for engineers and scienteists part 44

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Let rj, b, and c be the lengths of the sides of a triangle; let Q, /?, and 7 be the respective opposite angles (Fig. 3Aa): let B and r be the circumradius and the inradius, respectively; and let p = -j(a +h + c) be the semiperimeter. | 6.3. Functions of Several Variables. Partial Derivatives 269 6.3.4. Extremal Points of Functions of Several Variables 6.3.4-1. Conditions of extremum of a function of two variables. 1 . Points of minimum maximum or extremum. A point x0 y0 is called a point of local minimum resp. maximum of a function z f x y if there is a neighborhood of xo yo in which the function is defined and satisfies the inequality f x y f x0 y0 resp. f x y f x0 yo . Points of maximum or minimum are called points of extremum. 2 . A necessary condition of extremum. If a function has the first partial derivatives at a point of its extremum these derivatives must vanish at that point. It follows that in order to find points of extremum of such a function z f x y one should find solutions of the system of equations fx x y 0 fy x y 0. The points whose coordinates satisfy this system are called stationary points. Any point of extremum of a differentiable function is its stationary point but not every stationary point is a point of its extremum. 3 . Sufficient conditions of extremum are used for the identification of points of extremum among stationary points. Some conditions of this type are given below. Suppose that the function z f x y has continuous second derivatives at a stationary point. Let us calculate the value of the determinant at that point A fxxfyy - fXy. The following implications hold 1 If A 0 fxx 0 then the stationary point is a point of local minimum 2 If A 0 fxx 0 then the stationary point is a point of local maximum 3 If A 0 then the stationary point cannot be a point of extremum. In the degenerate case A 0 a more delicate analysis of a stationary point is required. In this case a stationary point may happen to be a point of extremum and maybe not. Remark. In order to find points of extremum one should check not only stationary points but also points at which the first derivatives do not exist or are infinite. 4 . The smallest and the largest values of a function. Let f x y be a