[delta]²u [delta]²u / [delta]²u \² ---------- · ---------- = ( ------------------- ) [delta]x1² [delta]x2² \ [delta]x1 [delta]x2 / then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of. Take for instance the function u = x² - xy² + y². Here the values x = 0, y = 0 satisfy the equations [delta]u/[delta]x = 0, [delta]u/[delta]y = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are ([delta]x)², and are never negative. Here [delta]u = [delta]x² - [delta]x[delta]y² + [delta]y², and by putting [delta]x = 0 or an infinitesimal of the same order as [delta]y², we can make the sign of [delta]u depend on that of [delta]y², and so be positive or negative as we please. On the other hand, if we take the function u = x² - xy² + y^4, x = 0, y = 0 make zero a critical value of u, and here [delta]u = [delta]x² - [delta]x[delta]y² + [delta]y^4, which is always positive, because we can write it as the sum of two squares, viz. ([delta]x - ½[delta]y²)² + ¾[delta]y^4; so that in this case zero is a minimum value of u. A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and [phi]´(x) = 0 has no roots in this interval, it follows that [phi]´(x) is one-signed as x increases from a to b, so that [phi](x) is increasing or diminishing all the time, and the greatest and least values of [phi](x) are [phi](a) and [phi](b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec [alpha]. Where should he land?