Derivative Question?

Question

If f'(x) has a cgabge of sign and is always defined, then f(x) has either a inflection point or a zero.

Is this statement true?

iam sorry there's a mistyped, it should be a change of sign

Answer 1

If f'(x) has a sign change and f(x) everywhere defined (we usually like to say 'continuous' ☺) then f(x) will certainly have an inflection point or a minimum or a maximum. I think you have the zero part confused. The minimum (or maximum or inflection point) occurs at the x value where f'(x) = 0. But the way you said it in the question implied that there was a point at which f(x) = 0 (which would make x a 'root' of the function) and, in general, this is not required.

And yes, if f(x) = k (the constant function) the f'(x) is everywhere 0.

OK?


Doug

Answer 2

What is a 'cgabge'? Do you mean, 'change of sign'? If so, where does it change sign?

Here is what I think you are saying:

f'(x) changes sign somewhere and f'(x) is defined everywhere, so, there is some point where f'(x) = 0.

This means that at this point,
f(x) has a local extremum or an inflection point.

Answer 3

That is not true.

f(x) = x^2 +5 ... f´(x) = 2x

You have f´(x) with a change of signal and is always defined, but the function f(x) has not a zero neither a inflection point.

Answer 4

if there's no variable the derivative is always 0

Answer 5

i don't know I'm confused