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Problem 77 Hard Difficulty

Prove that the function
$$ f(x) = x^101 + x^51 + x + 1 $$
has neither a local maximum nor a local minimum.

Answer

$f(x)=x^{101}+x^{51}+x+1 \Rightarrow f^{\prime}(x)=101 x^{100}+51 x^{50}+1 \geq 1$ for all $x,$ so $f^{\prime}(x)=0$ has no solution. Thus, $f(x)$
has no critical number, so $f(x)$ can have no local maximum or minimum.

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Video Transcript

Let's first take the derivative. As we can see, we have positive coefficients and we have powers that are even 150 or both even. Therefore, we know it's always increasing something that's always increasing. Let's say it looks like this or looks like this. It doesn't have a minimum of Maxim because it's always constantly increasing.