## anonymous one year ago Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x3 − 9x on the interval [−1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

1. anonymous

Basically the average function value has to be equal to the value of the function at some point on the interval. $\frac{ 1 }{ b-a } \int\limits_{a}^{b}f(x)dx=f(c)$ $\frac{ 1 }{ 1-(-1) } \int\limits_{-1}^{1}(x^3-9x)dx=c^3-9c$

2. anonymous

solve for c to get the values of the x-coordinates

3. thomas5267

The first mean value theorem for integration states If G : [a, b] → R is a continuous function and $$\varphi$$ is an integrable function that does not change sign on the interval (a, b), then there exists a number x in [a, b] such that $\int_a^b G(t)\varphi (t) \, dt=G(x) \int_a^b \varphi (t) \, dt.$ Take $$\varphi(t)=1$$ and you are done. Mean value theorem for integration applies for f(x). Find the x coordinate using peachpi's method. Copied shamelessly from Wikipedia.