Should be a fairly quick question, but I was given the following graph of $g(x)$:


which was followed by the question:

Let $h(x)=\int_{-4}^xg(x)\,dx$. On what open intervals contained in $-4\le x\le11$ is the graph of $h$ concave down? Give a reason for your answer.

So I'm a bit confused on how I'd find the values of $h(x)$, I'm assuming I just determine the respective integral for the specific value of $x$, like if I wanted to find $h(2)$ I'd plug 2 into the integral attached to $g(x)$ and I know that concavity is determined from the second derivative, but I'm extremely confused on how I'd go about finding the values of $h''(x)$. Any help is appreciated!

  • $\begingroup$ Is that graph of $g(x)$? $\endgroup$ – Parcly Taxel Apr 25 '20 at 1:22
  • $\begingroup$ Yeah, I forgot to include that, I'll put it in $\endgroup$ – joe Apr 25 '20 at 1:29

$h(x)$ is concave down at $x$ iff $h'(x)=g(x)$ is decreasing at $x$. We see that $g(x)$ is decreasing on $(1,4),(4,6)$ and $(8,11)$, so these are the same intervals on which $h(x)$ is concave down.


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