LOCATION OF THE ROOTS.
597. If the variable x changes continuously from a to b the function f(x) will change continuously from f(a) to f(b).
Let c and c+h be any two values of x lying between a and b. We have
and by taking h small enough, the difference between f(c+h)
and f(c) can be made as small as we please; hence to a small
change in the variable x there corresponds a small change in
the function f(x), and therefore as a changes gradually from
a to b, the function f(x) changes gradually from f(a) to f(b).
598. It is important to notice that we have not proved that f(x) always increases from f(a) to f(b), or decreases from f(a) to f(b), but that it passes from one value to the other without any sudden change; sometimes it may be increasing and at other times it may be decreasing.
599. If f(a) and f(b) are of contrary signs then one root of the equation f(x) = 0 must lie between a and b.
As x changes gradually from a to b, the function f(x) changes gradually from f(a) to f(b), and therefore must pass through all intermediate values; but since f(a) and f(b) have contrary signs the value zero must lie between them; that is, f(x) = 0 for some value of x between a and b.
It does not follow that f(x) = 0 has only one root between a and b; neither does it follow that if f(a) and f(b) have the same sign f(x) = 0 has no root between a and b.
600. Every equation of an odd degree has at least one real root whose sign is opposite to that of its last term.
In the function f(x) substitute for x the values +\infty, 0, -\infty, successively, then
f(+\infty)= +\infty, f(0)=p_n, f(-\infty)= -\infty
