Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots.

Question

Prove that: If $a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x=0$ has positive roots, then $na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+ a_{1}=0$ has positive roots. P/s: My grammar isn't good, so that my question is hard to understand. I’m sorry for the inconvenience

Answer 1

Let $\alpha$ be a positive root of the given function ($=f(x)$,say).We know that 0 is also a root of f(x). Apply, Rolle's theorem between them(We can apply that as polynomials are both continuous and differentiable on $\mathbb R$) $$\Longrightarrow \exists c\in(0,\alpha):f'(c) = 0$$ which is the required statement to be proven.

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