# How do you find the point c in the interval -4<=x<=6 such that f(c) is equation to the average value of f(x)=2x?

The average value of a function $f$ on interval $\left[a , b\right]$ is

$\frac{1}{b - a} {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$.

So we need to find the average value, then solve the equation $f \left(c\right) = \text{average value}$ on the interval $\left[a , b\right]$.

For our case we get

Solve:

$2 c = \frac{1}{6 - \left(- 4\right)} {\int}_{-} {4}^{6} \left(2 x\right) \mathrm{dx}$

Evaluating the integral we get:

Solve: $2 c = 2$.

The only solution is $c = 1$