Answer
x $\in$ (-1,1)
Work Step by Step
We need to find where the graph $F(x)$ is concave downward. By using the given information that $F(x)=\int_{1}^{x} f(t) dt$, we can determine that $F'(x)=f(t)$, and therefore $F''(x)=f'(t)$. To determine concavity, we must consider the second derivative.
By evaluating the given graph, we can see that $f'(t)$ changes sign when $x=-1$ and $x=1$, therefore $F(x)$ has two points of inflection (at -1 and 1).
The $+/-$ chart for $f'(t)$ can be found below, and it shows the graph is negative only between $x=-1$ and $x=1$, not inclusive. This tells us that $F''(x)$ is only concave downward from (-1,1).
