# Larson 1.11.5

Here’s a cute little problem: let $f(x)$ be a polynomial of degree $n$ with real coefficients such that it has non-negative values. Show that $S(x)= f(x) + \cdots + f^{(n)}(x) \ge 0$ for all real $x$.

Notice that $S(x)$ goes to positive infinity in both limits, as $f(x)$ is an even order polynomial. Furthermore, it is also non-negative at 0. Now, the minima/maxima of $S(x)$ exists where $f'(x) + \cdots + f^{(n)}(x) = 0$, but evaluating at such points reveals that the extremal values are all positive.

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