MAXIMA AND MINIMA. The maxima and minima of a function f(x) are the points of the curve where y is larger or smaller, respectively, than for any value of x differing from the value at that point by less than some finite number. According to this definition, a curve may have several maxima or minima, and a maximum or minimum need not be a point where y assumes the greatest or least value, respectively, throughout the entire curve. It is easy to see that a maximum is a point where the curve ceases to go up and begins to go down. At such a point, the slope dy of the curve, or the differential coefficient I dx changes sign. If is continuous at a maxi mum, it can only be 0, and at any rate, on an algebraic curve, it can only be 0 or co. Thus the maxima — likewise the minima — are all to be found among the points where dy --=-0 or CO dx have a minimum, < , a mum. However may be 0 or co out their being a maximum or minimum at dly the point. In thi; case — dx2 will be O. To determine whether we have a maximum, a mini mum or neither when higher deriva tives must be considered.
The definition of the maxima and minima of a function of two variables is obvious. It is clear that a necessary condition for a maximum or minimum of f(x, y), at a point where its partial derivatives are continuous, a af is = O. A sufficient condition for a maximum is ( ax a x. l and < o.
ay ax2 A sufficient conditionfora minimum is < axoy (l (l and > O.
ax ars Some important theorems in plane geometry having to do with maxima and minima are that of all polygons having a given perimeter and a given number of sides the regular one has the greatest area; that a circle ,has a greater area than any other figure with the same peri meter; that a right triangle is larger than any other with two sides equal to its legs, and that of all triangles with a given perimeter and a given base, the isosceles is the greatest. See Ou-cutus, THE INFINITESIMAL; CALCULUS OF VARIATIONS.