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

NWeid1

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


2. Relevant equations



3. The attempt at a solution
I think I should use the MVT so here is what I tried:

[tex]f'(c) = \frac{f(a) - f(0)}{a-0}[/tex]
f(0)=0 is given so:
[tex]f'(c) = \frac{f(a)}{a}[/tex]

Now I am confused on how to relate this to f(x)>0.

Joffan

Think about what f''(0)>0 means for f'(x) in the immediate vicinity of 0.

NWeid1

Does it mean that it is increasing? So f'(x)>0? Which means f(x) is increasing from f(0) which means f(x)>0 for a near 0?

Joffan

... for x in some region just greater than 0, yes.

LCKurtz

Assuming what you mean by "it is increasing" is "f'(x) is increasing near 0", yes. Good intuition, but of course, that is what you are supposed to prove.

Mark44

What you say will be clearer if you minimize the number of pronouns such as "it". The question involved f'' and f'. Which one of these do you mean by "it?"

NWeid1

Ok, I think I got it.

If a>0, and f''(0)>0 means f'(0) will be increasing so f'(x)>0 which means f(x) is increasing the origin and therefore f(x)>0. And since a>0 and f(a)>0

f'(c) = f(a)/a > 0
and therefore f(a) > 0. Is this right?

Mark44

This is not necessarily true. For example, if f(x) = x², f''(x) > 0 for all x, but the graph of y = f(x) is decreasing over half of its domain.

NWeid1

Yeah but since we're only looking at x>0, wouldn't it work? So confused, ugh! lol

Mark44

That was just an example. My point is that f''(x) being positive doesn't necessarily mean that f is increasing.

If you want an example where x > 0, consider f(x) = (x - 10)². f''(x) = 2 > 0, but there is an interval, namely [0, 10], on which the graph is decreasing.

NWeid1

Ok. I got it now. So now I'm confused. I see now that i used f(a)>0 and a>0 to prove that f(a)>0 lol....But, since f''(0)>0, f'(x) will be increasing at 0, right? And if f'(0)=0 and it is increasing, when x>0 for x near 0, f'(x)>0. Is this right at all? lol

Mark44

f'(x) will be increasing in some interval around 0. Increasing applies to an interval, not just a single point.Who is "it"?