Question

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`?

Answer 1

The average value of a function `f` on interval `[a,b]` is

`1/(b-a) int_a^b f(x) dx`.

So we need to find the average value, then solve the equation `f(c) = "average value"` on the interval `[a,b]`.

For our case we get

Solve:

`2c = 1/(6-(-4)) int_-4^6 (2x) dx`

Evaluating the integral we get:

Solve: `2c = 2`.

The only solution is `c=1`