Consider f be a real continuous function , $f(0) = 0$ , and whose graph has the form shown in the figure:
a) How can a give description of the function whose graph corresponds to Figure.
b) Sketch the graph of the integral $F(x)= \int_{0}^{x} f(s) ds$ in the corresponding space.
Graphic of f
Graphic of F
For a), I try to use the roots of $f, f ', f''$, and the signs of $f, f',f''$ but I get stuck finding the roots of $f$. For b), I could not do anything, please help.
Let's label some relevant $x$ values:
Now $F(x)=\int_0^x f(s)\,ds\implies F'(x)=f(x) \text{ and } F''(x)=f'(x)$ by the Fundamental Theorem of Calculus. Thus, from the graph,
$F'(x)=f(x)>0$ on $(-\infty,0)\cup(a,b)$ and so $F$ is increasing here.
$F'(x)=f(x)<0$ on $(0,a)\cup(b,\infty)$ and so $F$ is decreasing here.
Thus, $F$ has a local max at $x=0,b$ and local min at $x=a$.
$F''(x)=f'(x)>0$ on $(c,d)$ and so $F$ is concave up here.
$F''(x)=f'(x)>0$ on $(-\infty,c)\cup(d,\infty)$ and so $F$ is concave down here.
Thus, $F$ has inflection points at $x=c,d$.
Hopefully you can sketch a curve that satisfies all those conditions. Here's a (very rough!) candidate: