522 
DERIVATES, EXTREMES, VARIATION 
now the point 1) with the point 2), which we indicate as usual by 
We define now a function g(x) as follows : 
9<£) =/0) > when x ~ £. 
This defines g for all the points of (£. In the interval 3«? let g 
have a constant value. Obviously g is continuous, and has a 
pantactic set of intervals in each of which g is constant. 
3. We have given criteria for maxima and minima in I, 468 
seq., to which we may add the following: 
Let f{x) be continuous in (a— 5, a + 8). If Rf (a) > 0 and 
Lf'(a) < 0, finite or infinite, fix') has a minimum at x = a. 
If Lffia')<c 0 and Lf' (a)>0, finite or infinite, f (x) has a maxi 
mum at x = a. 
For on the 1° hypothesis, let us take a such that R — a> 0. 
Then there exists a 8’ > 0 such that 
Hence 
/(a + A)-/(a) >g _ g>0 t 0<A <g,. 
h 
/(a + A) >/(a) , a + h in (a*, a + S'). 
Similarly if /3 is chosen so that L + /3 < 0, there exists a 8" > 0, 
such that n, N 
/( g -A)-/(g) 
— h 
Hence 
f(a — h)>f(a) , a 4- A in (a — S", a*). 
520. Example 1. Let f(x) oscillate between the a;-axis and the 
two lines y = x and y = — x, similar to 
y = 
• 7T 
zsm— . 
x 
In any interval about the origin, y oscillates infinitely often, hav 
ing an infinite number of proper maxima and minima. At the 
point a: = 0,/has an improper minimum. 
Example 2. Let us take two parabolas _Pj, P 2 defined by y = x 2 , 
y = 2 x*. Through the points x=±^, let us erect ordi 
nates, and join the points of intersection with P 1 , P 2 , alternately 
by straight lines, getting a broken line oscillating between the