Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f''(x)>0$ for all $x\in \mathbb{R}$.Then which is true? Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f''(x)>0$ for all $x\in \mathbb{R}$.Then which  of the following is true?

1.$f(x)=0$ has exactly two solutions on $\mathbb{R}$.
2.$f(x)=0$ has a positive solution  if $f(0)=0$ and $f'(0)>0$.
3.$f(x)=0$ has no positive solution if $f(0)=0$ and $f'(0)>0$.
4.$f(x)=0$ has no positive solution if $f(0)=0$ and $f'(0)<0$.


My thoughts:-
(1) is not true as $f(x)=x^2+1$ is a counter example.
Now suppose the conditions in (2) holds.then $f'(x)$ is increasing everywhere.so $f'(x)$ is never zero for all positive $x$.so $f(x)$ can not have a positive solution otherwise $f'(x)$ have a zero between $0$ and $x$ by Rolle's Theorem.
so (3) is true.
Are my arguments right?

Correct answer for above question is option no.3.Second option is not correct because it has a counter example.$$f(x)=e^x-1$$ f satisfy condition $f(0)=0,f'(0)>0$but $0$ is the only solution of $f(x)=0$.Hence $f(x)=0$ has no positive solution. 
