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

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 $$f$$.

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.

• You are asking pretty good questions so far - keep it up! – Parcly Taxel Nov 5 '18 at 2:35

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.
• 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, $y=0$. – Ella Nov 5 '18 at 2:41
• @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. – Parcly Taxel Nov 5 '18 at 2:44