# . An elementary course of infinitesimal calculus . s derivative ^(x) are both of them continuous (andfinite) for all finite values of x. Hence at least one real rootof the equation (x) = 0 (1) 48-49] APPLICATIONS OF THE DERIVED FUNCTION. 105 will lie between any two real roots of «^(^) = 0 (2). This result, which is known as RoUes Theorem, is importantin the Theory of Equations. It is an immediate consequencethat at most one real root of (2) lies between any twoconsecutive roots of (1). That is, the roots of (1) separatethose of (2). JSx. 1. If &lt;^ (x) = 4a!» - 2la? + ISa; + 20,we have &lt;^

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. An elementary course of infinitesimal calculus . s derivative ^(x) are both of them continuous (andfinite) for all finite values of x. Hence at least one real rootof the equation (x) = 0 (1) 48-49] APPLICATIONS OF THE DERIVED FUNCTION. 105 will lie between any two real roots of «^(^) = 0 (2). This result, which is known as RoUes Theorem, is importantin the Theory of Equations. It is an immediate consequencethat at most one real root of (2) lies between any twoconsecutive roots of (1). That is, the roots of (1) separatethose of (2). JSx. 1. If &lt;^ (x) = 4a!» - 2la? + ISa; + 20, we have &lt;^ (x) = 120? - 42a! + 18 = 6 (2a; - 1) (a; - 3). Hence the real roots of &lt;^ (x) = 0, if any, will lie in the intervalsbetween — oo and J, ^ and 3, 3 and + oo , respectively. Now, for as = — 00 , J, 3, + 00, the signs of (x) are —, +, —, +, respectively, so that ij&gt; (x) must in fact vanish once (by Art. 10)in each of the above intervals. Hence there are three realroots. The figure shews the graph of (a!).. Fig. 33. If by continuous modification of the form of &lt;j){x), forexample by the addition or subtraction of a constant, tworoots are made to coalesce, the root of (x) = 0 which liesbetween must coalesce with them. Hence a double root of&lt;^(a!) = 0 is also a root of &lt;j)(x) = 0. 106 INFINITESIMAL CALCULUS. [CH. Ill More generally, an r-fold root of ^ (a;) = 0 being regardedas due to the coalescence of r distinct roots, the equation&lt;^ (x) = 0 will have r — 1 intent ening roots which coalesce. This suggests a method of ascertaining the multiple roots, if any, of a proposed algebraic equation. If a be an r-foldroot of &lt;^(a;), we have {x) = {x-arx{oo) (3). where % (x) is a rational integral function. Hence f {x) = {x- ay-^ [rx (x) + (x-a) x (ix:)} (4); i.e. (x — ay~^ will be a common factor of &lt;^(«) and ^{x).And it- is easily seen that (x — a)^^ will not be a commonfactor unless ^ {x) is divisible by (x — ay. Hence themultiple roots of ^ (x), if

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