Even degree polynomial.
By Sophia Bowman
I am trying to prove that if $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdot\cdot\cdot+a_{0}$ is an even degree polynomial with $a_{n}>0$ then its range is of the form $[k,\infty).$
Since $\lim p(x)=\infty$ as x tends to $\infty $ or $-\infty $ and $p(x)$ is continuous its range is bounded below. Its range must be of the form $(k,\infty)$ or $[k,\infty).$ How do I discard the first case?
$\endgroup$ 11 Answer
$\begingroup$There is a positive real number $B_1$ such that $p(x)\gt k$ for all $x\gt B_1$.
There is a positive real number $-B_2$ such that $p(x)\gt k$ for all $x\lt -B_2$.
On the closed interval $[-B_2,B_1]$ the continuous function $p(x)$ attains a minimum.
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