Let $f, g :\left[-1, 2\right]\to \mathbb{ }\mathbb{R}$   be continuous functions which are twice differentiable on the interval $\left(-1, 2\right)$. Let the values of $f$ and $g$ at the points $-1, 0$ and $2$ be as given in the following table:
 

	$x= -1$	$x = 0$	$x = 2$
$f\left(x\right)$	3	6	0
$g\left(x\right)$	0	1	-1

In each of the intervals $\left(-1, 0\right)$ and $\left(0, 2\right)$ the function $(f-3g)"$ never vanishes. Then the correct statement(s) is(are)

• A

  $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly three solutions in $\left(-1, 0\right) \cup (0, 2)$
• Correct answer
  B

  $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in (-1, 0)
• Correct answer
  C

  $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly one solution in (0, 2)
• D

  $f^{\prime }\left(x\right)-3g^{\prime }\left(x\right)=0$ has exactly two solutions in (-1, 0) and exactly two solutions in (0, 2)

Hence, the correct answer is options: 'B', 'C'.