Paragraph: Consider the function $f(x)=3 x^4-20 x^3-12 x^2+288 x+1$
Question: In which one of the following intervals is the function increasing?

• Correct answer
  A

  $(-2,3)$
• B

  $(3,4)$
• C

  $(-3,-2)$
• D

  $(-4,-3)$

Solution

$f(x)=3 x^4-20 x^3-12 x^2+288 x+1$ $\therefore f^{\prime}(x)=12 x^3-60 x^2-24 x+288$ $\begin{aligned} & \Rightarrow f^{\prime}(x)=(x+2)\left(x^{\prime}-3\right)(x-4) \\ & f^{\prime}(x) \gt 0 \\ & \Rightarrow(x+2)(x-3)(x-4) \gt 0\end{aligned}$
$\therefore \quad x \in(-2,3) \cup(4, \infty)$

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