If the equation a_n x^n + a_{n-1} x^{n-1} + d | vedclass

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(D) Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x$.Since $f(0) = 0$ and $f(\alpha) = 0$,where $\alpha > 0$,the function $f(x)$ satisfies the c

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Smaller than $\alpha$

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(D) Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x$.Since $f(0) = 0$ and $f(\alpha) = 0$,where $\alpha > 0$,the function $f(x)$ satisfies the conditions of Rolle's Theorem on the interval $[0, \alpha]$.According to Rolle's Theorem,there exists at least one root of the derivative $f'(x) = 0$ in the open interval $(0, \alpha)$.The derivative is $f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1$.Therefore,the equation $n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1 = 0$ must have at least one positive root smaller than $\alpha$.

If the equation $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x = 0$,where $a_1 \neq 0$ and $n \ge 2$,has a positive root $x = \alpha$,then the equation $n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1 = 0$ has a positive root,which is:

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