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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