Answer
A
Work Step by Step
The average rate of change of $f(x)$ over the interval $[a, b]$ is
$\displaystyle \frac{Change\ in\ f}{Change\ in\ x}=\frac{\Delta f}{\Delta x}=\frac{f(b)-f(a)}{b-a}$
$=$ Slope of line through pcints $P(a,f(a))$ and $Q(b,f(b))$
... $f^{\prime}$ associates to each $x$ the slope of the tangent to the graph of the function f at $x$,
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Reading from the graph points (2,0) and (4,1)
the average rate of change of $f$ over [2,4] is
$\displaystyle \frac{f(4)-f(2)}{4-2}=\frac{1-0}{4-2}=\frac{1}{2}.$
Also, at x=2, the tangent to the graph is horizontal, (slope =0), so$ f^{\prime}(2)=0$
$\displaystyle \frac{1}{2}$ is greater than 0, so the correct choice is A.
