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There is a calculus question that I read which doesn't seem meaningful, and it reads like this:

Find the absolute maximum and the absolute minimum of the function

$$f(x) = 9x^2 + 3x^3 + 40x$$

The question does not specify an interval on the real line to consider — is such a question meaningful?

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    $\begingroup$ Can the answer be "there is no absolute maximum and no absolute minimum"? $\endgroup$ Commented Nov 19, 2017 at 9:26
  • $\begingroup$ Hi @ArturoMagidin -- I would think that is the only answer, but I do wonder whether the question was meant to be deliberately asked without specifying an interval, as a teaching tool, or whether it was a bad question. $\endgroup$ Commented Nov 19, 2017 at 9:27

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If the answer assumes that the absolute maximum and the absolute minimum exist, then it is not meaningful. But if saying that there is no absolute maximum nor absolute minimum is an acceptable answer, then, yes, it is meaningful.

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Assuming we work over $\mathbb R$, it is :

$$f'(x) = 6x^2 + 18x + 40$$

and : $$Δ=18^2 - 4\cdot40\cdot6= 324 - 960 <0$$

which means that $f'$ sustains its sign.

For $x=0$ you can easily get that : $f'(0) = 40 > 0$, which, due to the above, means that $f'(x) > 0 \forall x\in \mathbb R.$ This means that the function $f$ is strictly increasing and thus does not have a minima or a maxima, which means that :

If the answer can be : there is no absolute minimum or maximum value then the question is meaningful, otherwise, it's not.

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  • $\begingroup$ Isn't it easier to note that $\lim_{x\to\infty}f(x) = \infty$ and $\lim_{x\to-\infty}f(x) = -\infty$ ? $\endgroup$ Commented Nov 20, 2017 at 0:30

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