Properties of a twice-differentiable function

[Hitman2-2]

Homework Statement

Suppose that f is a twice-differentiable function on the set of real numbers and that f(0), f'(0), and f''(0) are all negative. Suppose f'' has all three of the following properties.

I. It is increasing on the interval $$[0, \infty)$$.
II. It has a unique zero in the interval $$[0, \infty)$$.
III. It is unbounded on the interval $$[0, \infty)$$.

Which of the same three properties does f necessarily have?

Homework Equations

The Attempt at a Solution

I thought that f would have all three properties, but the answer is that it only has properties II & III and I don't see why. If f isn't increasing on the interval, then wouldn't it have to eventually decrease without bound? It looks to me that the first and second derivatives are increasing without bound so why isn't this a contradiction?

 

[olgranpappy]

Homework Statement

Suppose that f is a twice-differentiable function on the set of real numbers and that f(0), f'(0), and f''(0) are all negative. Suppose f'' has all three of the following properties.

I. It is increasing on the interval $$[0, \infty)$$.
II. It has a unique zero in the interval $$[0, \infty)$$.
III. It is unbounded on the interval $$[0, \infty)$$.

Which of the same three properties does f necessarily have?

Homework Equations

The Attempt at a Solution

I thought that f would have all three properties, but the answer is that it only has properties II & III and I don't see why. If f isn't increasing on the interval, then wouldn't it have to eventually decrease without bound? It looks to me that the first and second derivatives are increasing without bound so why isn't this a contradiction?

The slope of f is initially negative, so it has to decrease initially.

 

[Hitman2-2]

Well, I totally overlooked that.

Thanks for the help.