The equation frac{3}{x - a^3} + frac{5}{x - a | vedclass

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Question

(A) Let $f(x) = \frac{3}{x - a^3} + \frac{5}{x - a^5} + \frac{7}{x - a^7}$.Given $a > 1$,we have $a^3 < a^5 < a^7$.As $x 	o (a^3)^+$,$f(x) 	o +\inft

Vedclass · Accepted Answer

Two real and positive roots

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(A) Let $f(x) = \frac{3}{x - a^3} + \frac{5}{x - a^5} + \frac{7}{x - a^7}$.Given $a > 1$,we have $a^3 As $x 	o (a^3)^+$,$f(x) 	o +\infty$. As $x 	o (a^5)^-$,$f(x) 	o -\infty$. Thus,there is a root in $(a^3, a^5)$.As $x 	o (a^5)^+$,$f(x) 	o +\infty$. As $x 	o (a^7)^-$,$f(x) 	o -\infty$. Thus,there is a root in $(a^5, a^7)$.Since $a > 1$,all values in the intervals $(a^3, a^5)$ and $(a^5, a^7)$ are positive.Therefore,the equation has two real and positive roots.

The equation $\frac{3}{x - a^3} + \frac{5}{x - a^5} + \frac{7}{x - a^7} = 0$,where $a > 1$,has:

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