Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a differentiable function such that $f^{\prime}(0)=1$ and $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$.
Which of the following is true?

• A

  $f$ is a decreasing function but $f^{\prime}$ is an increasing function.
• Correct answer
  B

  Both $f$ and $f^{\prime}$ are increasing functions.
• C

  $f$ is an increasing function but $f^{\prime}$ is a decreasing function.
• D

  Both $f$ and $f^{\prime}$ are decreasing functions.

Hence, the correct answer is option: 'B'.