Integral inequality: Find "xi" in (a,b) so |f'(xi)| >= [1/(b-a)&^2] int[a,b]|f(x)| dx

[kopanagiotou]

Integral inequality: Find "xi" in (a,b) so |f'(xi)| >= [1/(b-a)&^2] int[a,b]|f(x)| dx

1) f (x) is continuous in [a, b] and differentiable in (a, b).

2) f (a) = f (b) = 0

Under the two assumptions listed above, prove that there exists some \(\displaystyle \, \xi\, \in\, (a,\, b)\, \) such that the following holds:

. . . . .\(\displaystyle \bigg|\, f'(\xi)\, \bigg|\, \geq\, \dfrac{4}{(b\, -\, a)^2}\, \)\(\displaystyle \displaystyle \int_a^b\, \bigg|\, f(x)\, \bigg|\, dx\)

 

[stapel]

1) f (x) is continuous in [a, b] and differentiable in (a, b).

2) f (a) = f (b) = 0

Under the two assumptions listed above, prove that there exists some \(\displaystyle \, \xi\, \in\, (a,\, b)\, \) such that the following holds:

. . . . .\(\displaystyle \bigg|\, f'(\xi)\, \bigg|\, \geq\, \dfrac{4}{(b\, -\, a)^2}\, \)\(\displaystyle \displaystyle \int_a^b\, \bigg|\, f(x)\, \bigg|\, dx\)

What are your thoughts? What have you tried? Where are you stuck?

For instance, the "assumptions" might put one in mind of Rolle's Theorem (here) or the Mean Value Theorem. The absolute values might put one in mind of some of the properties of integrals (here). what did you come up with?

By the way, are you supposed to be taking, for \(\displaystyle \,\xi,\,\) the x-value that gives the maximum value of f' on the interval (like page 5 here)?

When you reply, please be complete. Thank you!

 

[kopanagiotou]

What are your thoughts? What have you tried? Where are you stuck?

For instance, the "assumptions" might put one in mind of Rolle's Theorem (here) or the Mean Value Theorem. The absolute values might put one in mind of some of the properties of integrals (here). what did you come up with?

By the way, are you supposed to be taking, for \(\displaystyle \,\xi,\,\) the x-value that gives the maximum value of f' on the interval (like page 5 here)?

When you reply, please be complete. Thank you!