Q. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0$ and $f^{\prime}(x)+2 f(x)>0$ for all $x \in \mathbb{R}$, where $f^{\prime}$ denotes the derivative of $f$ and $\mathbb{R}$ denotes the set of all real numbers. Then which one of the following statements is true?

A. $f(x)>0$, for all $x>0$ and $f(x)<0$, for all $x<0$

B. $f(x)<0$, for all $x \neq 0$

C. $f(x)>0$, for all $x \neq 0$

D. $f(x)<0$, for all $x>0$ and $f(x)>0$, for all $x<0$

It seems to me that option $A$ is true by taking the derivative of the function $g(x)=e^{2x}f(x)$, But I am not able to take counter examples for other options. Can we suggest some function $f(x)$ satisying the hypothesis.