# Same number of roots for derivative as function

1. Jan 12, 2014

### Persimmon

1. The problem statement, all variables and given/known data

Provide an example of a function such that f(x) has two and only two real roots and f'(x) has two and only two real roots, where f is defined for all real numbers and differentiable everywhere on its domain.

2. Relevant equations

3. The attempt at a solution

I know that if a function has n roots, it's derivative has to have at least n-1 roots. But I can't for the life of me visualize a function that would have two zeroes and have it's derivative also have exactly two zeroes. If anyone could give me a hint I'd be super thankful.

I don't know if this is enough of an attempt within the community guidelines but I'm really stuck.

2. Jan 12, 2014

### Dick

Did you try to sketch a graph that works? I don't think that's so hard. Once you've got that, if you are having a hard time writing an explicit function that looks like that you could always resort to defining it piecewise, as long as the function and the derivative are continuous where the pieces join up.

3. Jan 12, 2014

### Persimmon

Thanks, it seems really obvious to me now. D'oh!