Question

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.

Answer 1

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

Answer 2

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

Answer 3

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.

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