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

• Dec 7th 2011, 06:27 PM
NWeid1
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.
• Dec 7th 2011, 07:42 PM
Prove It
Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...
Quote:

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.
• Dec 8th 2011, 07:36 AM
Plato
Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...
Quote:

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 $\displaystyle \left( {\forall x \in (0,\delta )} \right)\left[ {f'(x) > x\frac{{f''(0)}}{2}} \right]$.
• Dec 8th 2011, 08:10 AM
NWeid1
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 :\
• Dec 8th 2011, 08:15 AM
Plato
Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...
Quote:

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?
• Dec 8th 2011, 08:16 AM
NWeid1
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