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Where is the graph of f(x) simultaneously increasing and concave down?

Ok, so I know that the answer is (-3,-2)U(1,2) but I don't know how you're supposed to get that answer. I've attached a picture of the graph.Math problem

Can someone please guide me and explain to me the process of solving this problem? Thank you!

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$f$ is increasing $\iff$ $f'$ is positive

$f$ is concave down $\iff$ $f''$ is negative.

We don't have a picture of $f''$, but since $f''$ is the derivative of $f'$, we know that

$f''$ is negative $\iff$ $f'$ is decreasing.

Therefore, we are looking for the places where $f'$ is positive and decreasing, which you can find from the picture.

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The answer is $x\in(-3,-2)\cup(1,2)$. This is because

  • The $f^\prime$ is positive on this interval, AND

  • $f^{\prime\prime}<0$, i.e., $f^\prime$ has negative slope.

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