Answer
Please see explanations below.
Graphs:
.
Work Step by Step
(Graphed with desmos.com)
Row 1, graph 1:
$f(x)$, zoomed out sufficiently to roughly see all areas "of interest".
Row 1, graph $2$:
We graph the derivative; note the zeros of $f'(x)$, and the intervals in which $f'(x)$ is positive or negative. From these data, we determine where $f(x)$ rises/falls, and where the local extremes lie.
$\left[\begin{array}{llllllllll}
& -\infty & & 0 & & 0.518 & & 3.992 & & \infty\\
f'(x) & & + & 0 & - & 0 & + & 0 & - & \\
f(x) & & \nearrow & max & \searrow & min & \nearrow & max & \searrow &
\end{array}\right]$
Row 1, graph 3:
combined graphs of $f(x)$ and $f'(x)$, with points of extrema.
Row 2:
graphs of $f''(x)$ and $f(x)$, with vertical lines where $f''$ is zero.
This gives us inflection points at $x=-0.117 $and $3.046$.
$\left[\begin{array}{lllllll}
& -\infty & & -0.117 & & 3.046 & \infty\\
f''(x) & & - & 0 & + & 0 & -\\
f(x) & & \cap & infl & \cup & infl & \cap
\end{array}\right]$
The graph in row 2 also gives points of inflection.
Row 3:
Zoom-in around the local minimum at $x=0.518$.
