# Thread: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

1. ## Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a). Can anything be concluded about f(x) for negative x's? My teacher hinted at using the Mean value theorem but I can never seem to get MVT problems right.

2. ## Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Originally Posted by NWeid1
Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a). Can anything be concluded about f(x) for negative x's? My teacher hinted at using the Mean value theorem but I can never seem to get MVT problems right.
The conditions given imply that there is a minimum at the origin. So that means that to the right of the origin, the function must increase, even if just for a little while.

3. ## Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Originally Posted by NWeid1
Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a).
Use the limit definition of derivative.
We know that $\displaystyle f"(0)$ exists and is positive.
Therefore, $\displaystyle \left( {\exists \delta > 0} \right)\left( {\forall x \in (0,\delta)}\right)\left[ {\left| {\frac{{f'(x) - f'(0)}}{{x - 0}} - f''(0)} \right| < \frac{{f''(0)}}{2}} \right]$.

From which we get $\displaystyle \left( {\forall x \in (0,\delta )} \right)\left[ {f'(x) > x\frac{{f''(0)}}{2}} \right]$.

4. ## Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Plato: I haven't learned this method :\

5. ## Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Originally Posted by NWeid1
Plato: I haven't learned this method :\
Are you saying that you never learned the definition of a derivative?
How can you be asked to do such a question then?

6. ## Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

Not the formal definition. My teacher said to use mean value theorem