# Maxima

## maximum, negative, minimum, positive, dy and differential

MAXIMA and MINIMA. These Latin words, which simply mean " greatest" and " least," are used to imply, not the absolute greatest and least values of a varying quantity, but the values which it has at the moment when it ceases to increase and begins to decrease, or vice versa. Thus, if it be said that the height of the barometer was a maximum at ten o'clock, it means that up to that hour the barometer rose, and then began to fall ; in which case it would still be said to have been a maximum, even though it should afterwards rise, and stand at a greater height than that at ten o'clock. Thus it is possible that there should be several maxima and minima in one day, and even that one of the minima should be greater than one of the maxima ; that is, at one moment when the fall ceases and a rise begins, the barometer may then be higher than it was at another time when a rise had ceased and a fall began.

The theory of maxima and minima is, mathematically speaking, very simple. It is obvious, from the definition of a differential coefficient, that if y be a function of x, and if x be increasing, then when y also increases, d dy x is positive; and when y diminishes, dy is negative. If the words increase and diminution have their full alge braical sense, this proposition is true whatever the sign of y may be. It follows that when increase ceases and diminution begins, d dy  changes x from positive to negative, and when diminution ceases and increase begins, it changes from negative to positive. But as a quantity cannot change its sign without becoming either nothing or infinite, it follows, first, that y can only be a maximum when x has such a value that dy is nothing or infinite ; secondly, that there is not then a maximum unless the latter change from positive to negative, when x increases through that value;' nor a minimum unless the same differential coefficient change from negative to positive, in the same case.

Thus when y=a +xx-, the differential coefficient of which is I-2x, we see that the latter changes sign when x changes from less than & to greater than 4; and the change of sign is from positive to negative. There is therefore a maximum when x=4, and this

maximum is a+ or a +i.

When dy  =0 (which is by far the most common case), and there is dz a maximum, it changes sign from + to , or diminishes, algebraically speaking : therefore dx2 is negative. 'Similarly, when = 0, and viz!, dy there is a minimum,  is positive. But when is infinite, there is a maximum or minimum, this additional rule does not apply.

Works on the differential calculus give the development of this theory and examples. We shall here only add one of the rules for determining the maximum or minimum when there are two distinct variables.

When z is a function both of x and y, two variables independent of dz dz one another, there may be a maximum or minimum when and dy are both nothing, both infinite, or one nothing and the other infinite. When they are both nothing, which is the only case in which this theory is of any practical application, it must be determined as follows whether there be any maximum or minimum, and which it is. FinA dz dz the values of x and with any pair  of these values find the value of the expression If this be negative, or nothing, there is a maximum or minimum ; if it be positive, there is a mixture of the two which can only be satis factorily explained by illustrations drawn from the theory of curved surfaces. When the expression is negative or nothing, there is a cl:z maximum if  and  be both negative, and a minimum if they be d.e both positive.

The usual method of establishing all the preceding formulae, namely, by the application of Taylor's theorem, applies only to the cases in which the differential coefficients become nothing, and not to that in which they become infinite. It is also frequently stated that there is always a maximum or minimum when a differential coefficient vanishes, which is not true.