588 VIII. ON THE STRUCTURE OF MATHEMATICS 

2. MAXIMA AND MINIMA
It will be useful to have some applications of the differential calculus explained.
If a function is continuous in an intervaland has larger
(or smaller) values at some intermediate points than it has at or near the ends, then it has a maximum (or minimum) at some pointinside this interval.
If Fig. 7 represents the graph of the function, it is obvious that at the maximum (or minimum) the tangent to the curve is parallel to the axis and therefore the slope of this tangent is zero. As this slope is given by the derivative and the slope is zero we have a simple method of finding the maximum (or the minimum) of a function by equating the first derivative to zero; namely,
It is useful to be able to discriminate between the maximum and the minimum of a function. Fig. 7 shows that this can be done by finding means to discriminate between the two cases when our curve is concave upwards or concave downwards. The slope of a curve for a particular value of x
is given by the value of , corresponding to that value of x. If the value D_{x}y is positive, y increases as x increases, and the curve slopes up as we move to the right; if the value of is negative,
y decreases as * increases, and the curve slopes down as we move to the right.




If we consider the curve y =/(*) which has its concave side turned upward (Fig. 8), the slope of the curve itself is a function of x, tan If we consider a variable point P on a curvetogether with the tangent
to the curve P, as following the curve in the direction of increasing values of x, the curve is concave upward whenever the slope is increasing algebraically, 
