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Recently I was reading through some practice problems for my linear algebra class and a question said we are allowed to assume the following.

If $f$ is a non-negative polynomial function on $[−1,1]$ and if $f(a)>0$ for some $a∈[−1,1]$, then $\int_{-1}^{1} f(t)dt >0$.

In the above statement, what is meant by non-negative polynomial? For reference we are working in the vector space of all polynomials of at most degree $n$.

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  • $\begingroup$ It probably means $f(x) \geq 0$ for all $x \in [-1,1]$ $\endgroup$ – D_S Nov 11 '17 at 22:44
  • $\begingroup$ I would assume that a non-negative polynomial is a polynomial that only takes non-negative values. If, say, we are working over $\mathbb R$, then $x^2$ is non-negative. $\endgroup$ – lulu Nov 11 '17 at 22:44
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The key is that it says $f$ is a non-negative function on $[-1, 1]$. So it just means $f(t) \geq 0$ for $t \in [-1, 1]$.

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