$$P(x)= x^n+ \sum\limits_{k=0}^{n-2} a_kx^k $$
Prove: $\exists i \in [1,n] \ $ and $i$ is an integer number such that $\;P(i)≥ \dfrac{n!}{\binom ni}$.
Prove that $\exists i \overline{1;n}$ such that $P(i)\,\geq\, \frac{n!}{\binom n2}$
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$\begingroup$
inequality
polynomials
combinations
factorial
lagrange-interpolation
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0By "$i= \overline{1;n}$", do you mean that $1\leq i \leq n$? – 2017-01-18
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0yes and also i is an integer number too – 2017-01-18
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2Any assumptions on the coefficients $a_k$? Because it's surely not true the way it's currently stated, without any restrictions on the coefficients. – 2017-01-18
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0there is no more about the conditions and the problem is also right too, I do assure about that – 2017-01-18
1 Answers
1
The proposition does not hold true as stated.
Consider for example $P(x) = x^n - a_0$ with $a_0 = n^n$. Then $P(i) \le 0$ for $1 \le i \le n\,$, so $P$ takes no positive values across that range, and so $P(i) \lt n! / \binom{n}{2}\,$ for $\forall i \in [1,n]\,$.