An introduction to the modern theory of equations, by Florian Cajori.

LOCATION OF THE ItOOTS OF AN I]QUATION

49

45. Rolle's Theorem. Between two soccessive real roots a and.)
b of the equation f(x)= 0 there lies at least one real root of the
equation f' (x) = 0.
Let the curve in this figure be the graph of f(x) = 0. The
points A, B, C,, E, F, G represent maxilmum and minitmum
values of f(x); the points if, N, Prepresent real roots of /(x) =0.
Between the two roots 1M and Nthe curve bends down and
F
A
D?
M N p
then up. Between the real root at N and the double root at
P the curve goes up, down, up, and finally down. Evidently,
between each pair of distinct successive real roots there must
be at least one maximum or minimum value off(x).
But each maximum or minimum point represents a value of
x which is a root of the equation f'(x) _0 (~ 44). Hence
Rolle's Theorem is proved.
From the examination of the figure we see that two successive
roots of the derived function may not comprise between them
any real root off(x) = 0, as in case of the roots represented by
D and E; they may comprise one dlistinct root, as in case of
the roots at A and B, B and C, E and F, but they can never
comprise more than one root of f(x) = 0.
E