I want to prove that a differentiable function $f: [0,\infty) \rightarrow R $, which satisfies the following condition is always non-negative:

Assume $f(0)=0$ and whenever $f(a)=0$, then $f'(a) \geq 0$ and $f''(a) \neq 0$.

Note that if I change the condition to this: $f(0)=0$ and whenever $f(a)=0$, then $f'(a) > 0$, then it's always non-negative. But in the condition I have $f'(a) \geq 0$ and that makes the problem a bit more complicated. Therefore I have added the second derivative condition $f''(a) \neq 0$.

So if this condition doesn't make it non-negative, what could be a slight modification on the condition which makes it non-negative? (e.g. $f'(a) \geq 0$ and $f''(a) > 0 ?$)

I want to prove that a differentiable function $f: [0,\infty) \rightarrow \mathbb{R} $, which satisfies the following condition is always non-negative:

Assume $f(0)=0$ and whenever $f(a)=0$, then $f'(a) \geq 0$ and $f''(a) \neq 0$.

Note that if I change the condition to this: $f(0)=0$ and whenever $f(a)=0$, then $f'(a) > 0$, then it's always non-negative. But in the condition I have $f'(a) \geq 0$ and that makes the problem a bit more complicated. Therefore I have added the second derivative condition $f''(a) \neq 0$.

So if this condition doesn't make it non-negative, what could be a slight modification on the condition which makes it non-negative? (e.g. $f'(a) \geq 0$ and $f''(a) > 0 ?$)