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478 Rh number of variations when x passes through a value which makes a function except f(x) vanish.

(2) Let c be a root of the equation f(x)=0 so that f(c) =0. Let h be any positive quantity.

Now f(c+h)=f(c)+ Mile i5 RO+ [Art. 594.]

and as c is a root of the equation f(x)=0, f(c)=0, hence ,

2 e+ N= MOF BHO +

If h be taken very small, we may disregard the terms containing its higher powers and obtain

SE + R= MO,

and as h is a positive quantity, f(c +h) and f_1(c) have the same sign. That is, the function just after x passes a root has the same sign as f_1(x) at a root.

In a like manner we may show that f(c — h)=— hf_1(c), or that the function just before c passes a root has a sign opposite to f(x) at a root. Thus as x increases, Sturm’s Functions lose one variation of sign only when x passes through a root of the equation f(x)= 0.

There is at no time a gain in the number of variations of sign, hence the theorem is established.

605. In determining the whole number of real roots of an equation f(x)= 0 we first substitute — \infty and then +\infty for x in Sturm’s Functions: the difference in the number of variations of sign in the two cases gives the whole number of real roots.

By substituting -\infty and 0 for x we may determine the number of negative real roots, and the substitution of +\infty and 0 for w gives the number of positive real roots.

606. When +\infty or —\infty is substituted for x, the sign of any function will be that of the highest power of x in that function.