Is there a nice way to prove that $f(x)=x^3+x^2+x-3$ is strictly increasing without making use of derivatives or any other advanced concepts ? I'm trying to explain it to a 9th grader, but I can't find an elegant, clear solution . Thanks !

  • $\begingroup$ Try explaining it visually by plotting a graph for each of the terms and in total? $\endgroup$ – hypergeometric Sep 6 '14 at 15:08
  • $\begingroup$ 9th graders aren't acquainted with function plotting, so I'm trying to avoid this method $\endgroup$ – Victor Sep 6 '14 at 15:09
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    $\begingroup$ $x^2$ isn't strictly increasing $\endgroup$ – Victor Sep 6 '14 at 15:35
  • $\begingroup$ @Victor Sorry, you're correct comment removed. I was only considering positive $x$. $\endgroup$ – Warren Hill Sep 6 '14 at 15:52

Explain $$f(x)-f(y)=(x-y)((x+y)^2+(x+1)^2+(y+1)^2)/2$$

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    $\begingroup$ Well, obviously $x-y$ is a factor, then we have $x^2+xy+y^2+x+y+1$ as the other factor. Now it is a matter of completing squares. This is simpler once you see it as $\frac12(x^2+2xy+y^2 \;+x^2+2x+1\;+y^2+2y+1)$. $\endgroup$ – Nemo Sep 6 '14 at 17:30

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