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Let $y = f(x)$ such that if $-0.5\le x$ then $y\le6.5$. The following diagram shows the graph of $f’$, the derivative of $f$.

diagram here

Find the set of values of $x$ for which the graph of $f$ is concave down.

On this graph, it’s concave down at (2,0), but this is the derivative of the actual function. On the actual function graph, $x=2$ would be a maximum. But I don’t know where to go from there.

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    $\begingroup$ You are asking pretty good questions so far - keep it up! $\endgroup$ – Parcly Taxel Nov 5 '18 at 2:35
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Remember that $$f(x)\text{ concave down}\iff f'(x)\text{ decreasing}$$ and we see that $f'(x)$ is decreasing only on $(2,4)$, so $f(x)$ is concave down only on that interval.

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  • $\begingroup$ This makes sense! But, I’m confused how you got 4 as the $y$ value in that point since when $x = 2$ on the graph, $y=0$. $\endgroup$ – Ella Nov 5 '18 at 2:41
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    $\begingroup$ @Ella 4 is the $x$ value, not the $y$ value. It's just eyeballing the graph of $f'(x)$. It reaches a local minimum at $x=4$ and a local maximum at $x=2$, so $f'(x)$ is decreasing between those points. $\endgroup$ – Parcly Taxel Nov 5 '18 at 2:44

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