#### Answer

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#### Work Step by Step

Prove that the function $f(x) = x^{101} + x^{51} + x + 1$ has no minimum or maximum values.
$f'(x) = 101x^{100} + 51x^{50} + 1$
Because $f'(x)$ is a polynomial with positive coefficients and even powers, it is always positive. Thus $f(x)$ is always increasing. If $f(x)$ is always increasing, it can have no maximum or minimum values.