Question

How do you find the numbers b such that the average value of `f(x) = 7 + 10x − 9x^2` on the interval [0, b] is equal to 8?

Answer 1

See the explanation below.

Explanation:

The average value of `f(x) = 7 + 10x − 9x^2` on the interval `[0, b]` is

`1/(b-0)int_0^b (7 + 10x − 9x^2) dx`

Which can be evaluated to,

`1/b[7x+5x^2-3x^3]_0^b = (7b+5b^2-3b^3)/b`

`= 7+5b-3b^2`

We want the average value to be `8`, so we need to solve

`7+5b-3b^2 = 8`.

Use the formula or complete the square to get

two solutions: `b=(5-sqrt13)/6` and `b=(5+sqrt13)/6`.

Since both are positive, the interval `[0,b]` exists for either of these.