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.
Best Answer
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.