# Mean Value Theorem : Roots of Derivatives on an Interval

Not what you're looking for? Search our solutions OR ask your own Custom question.

The function f (x) and all of its derivatives are continuous on [0, 10]. You know that f (0) = 0,

f (2) = 0, f (3) = 0, f (6) = 0, and f (8) = 0. At how many points must the first derivative of f (x)

be zero? At how many points must the second derivative of f (x) be zero? At how many points must the third derivative of f (x) be zero? And so on. Justify your answers.

https://brainmass.com/math/derivatives/mean-value-theorem-roots-derivatives-interval-163060

#### Solution Preview

We can use the mean value theorem.

f(0)=0, f(2)=0, f(3)=0, f(6)=0, f(8)=0

Then

1. For the first derivative, we can find x1 in (0,2), such that f'(x1)=0; we can find x2 in (2,3), such that f'(x2)=0;

we can find x3 in (3,6), such that f'(x3)=0; we can find ...

#### Solution Summary

Mean value theorem and roots of derivatives on an interval are investigated. The solution is detailed and well presented.

$2.49