Derivatives and Polynomial Functions

[mscbuck]

Homework Statement

Show that there is a polynomial function f of degree n such that:

1. f('x) = 0 for precisely n-1 numbers x
2. f'(x) = 0 for no x, if n is odd
3. f'(x) = 0 for exactly one x, if n is even
4. f'(x) = 0 for exactly k numbers, if n-k is odd

Homework Equations

The Attempt at a Solution

For the first one, I know with Rolle's Theorem that (x-1)(x-2)(x-3)...(x-n) would be a polynomial that worked since at x=1,2,3...n we have f(x) = 0, so that means between those intervals we must have a point where f'(x) = 0 up till [n-1, n], so that proves that.

I'm having trouble thinking about the other ones though. For #2, I was thinking that any odd function + a constant would work, but then I realized that it ignores possibility of f'(x) being negative + a constant = 0.

Any help is appreciated! Any hints at what kind of functions to think about? Thanks again!

 

[Mark44]

Don't overthink these.
For 2, what about f(x) = x?
For 3, can you think of a function whose graph has only one point where the tangent is horizontal.
For 4, starty by chosing a value of n, and choose a value of k so that n - k is odd. A good start might be n = 2 and k = 1.

 

[mscbuck]

Thanks a lot Mark44. I find my problem so far in this intro to analysis class is simply that I often approach problems the wrong way, or drastically over think them (like in this case!).

I was able to figure out 3 and 4 from your hints, thanks a lot!

 

[Mark44]

Maybe this problem will help you realize that problems can sometimes have simple solutions. When you're looking for possible approaches, consider the simplest first (that have a chance of succeeding).