# Question #c15cb

Mar 26, 2016

Any pair where $f \left(6\right)$ is $6$ greater than $f \left(3\right)$.

#### Explanation:

The average rate of change of the function $f \left(x\right)$ on the interval from $\left[a , b\right]$ is equivalent to

$\frac{f \left(b\right) - f \left(a\right)}{b - a}$

Since we know this is equal to $2$, and that our interval is $\left[3 , 6\right]$, we can say that

$\frac{f \left(6\right) - f \left(3\right)}{6 - 3} = 2$

$\frac{f \left(6\right) - f \left(3\right)}{3} = 2$

$f \left(6\right) - f \left(3\right) = 6$

Thus, any pair of values where $f \left(6\right)$ is $6$ greater than $f \left(3\right)$ will work. I can't see your answer options, but the pair might be something like:

$\left\{\begin{matrix}f \left(3\right) = 0 \\ f \left(6\right) = 6\end{matrix}\right.$

You could also have any of the following pairs. (Remember, the only thing that must hold true is that $f \left(6\right) = 6 + f \left(3\right)$.)

$\left\{\begin{matrix}f \left(3\right) = - 2 \\ f \left(6\right) = 4\end{matrix}\right.$

$\left\{\begin{matrix}f \left(3\right) = \frac{1}{2} \\ f \left(6\right) = \frac{13}{2}\end{matrix}\right.$

$\left\{\begin{matrix}f \left(3\right) = - 200 \\ f \left(6\right) = - 194\end{matrix}\right.$

$\left\{\begin{matrix}f \left(3\right) = \pi \\ f \left(6\right) = \pi + 6\end{matrix}\right.$