Let f(x) be a function with n distinct zeros on [a, b]. Prove that, f'(x) has at least n − 1 distinct zeros on [a, b].
Don't really know where to start here, any help is appreciated.
yes the function is differentiable on {a b} and continuous. We need to prove that there is n-1 zeros and not that there is a zero(rolle's) --> f'(c)= 0. For example the function itself may have 5 zeros on [a b] so its derivative will have 4 on [a b], that what is required to prove
Hello, somestudent2!
I must assume that the function is continuous on
Let be a function with distinct zeros on
Prove that has at least distinct zeros on
Maybe a sketch will help you visualize the problem . . .
Suppose has four distinct zeros onCode:| | --*-- | * * --*-- | * * * * - - + - + - o - - - - - o - - - - o - - - -o- - - + - - | a * * * b | * --*-- |
Then there are at least three horizontal tangents on