Answer
See graph
$g$ is negative when $f$ is decreasing
$g$ is positive when $f$ is increasing
$g=0$ when $f$ has a turning point (relative maximum)
Work Step by Step
We are given the functions
$f(x)=2x-x^2$
$g(x)=\frac{f(x+0.01)-f(x)}{0.01}$.
Determine $g(x)$ by plugging $f(x)$ into $g(x)$:
$g(x)=\frac{[2(x+0.01)-(x+0.01)^2]-(2x-x^2)}{0.01}$
$=\frac{2x+0.02-x^2-0.02x-0.0001-2x+x^2}{0.01}$
$=\frac{-0.02x+0.0199}{0.01}$
$=-2x+1.99$
We graph both functions.
We notice:
$g$ is negative when $f$ is decreasing
$g$ is positive when $f$ is increasing
$g=0$ when $f$ has a turning point (relative maximum)
