Question

If $f\left(x\right)=x^3-6x^2-36x+2$ is decreasing function, then $x\in$

• A. $\left(6,\infty \right)$
• B. $\left(-\infty ,-2\right)$
• C. $\left(-2,6\right)$
• D. None of these

Answer 1

The correct option is C

$\left(-2,6\right)$

Explanation for the correct option.

Step- 1: Form an inequality:

A function $f\left(x\right)$ is decreasing when $f\text{'}\left(x\right)<0$.

The differentiation of the function $f\left(x\right)=x^3-6x^2-36x+2$ is given as:

$\begin{array}{rcl}f\text{'}\left(x\right)& =& 3x^2-6\times 2x-36\\ & =& 3x^2-12x-36\end{array}$

Now, as the function is decreasing so $f\text{'}\left(x\right)<0$ and so the inequality becomes:

$$\begin{gathered} 3x^2-12x-36<0 \\ \Rightarrow x^2-4x-12<0 \end{gathered}$$

Step- 2: Solve the inequality:

The inequality $x^2-4x-12<0$ can be solved as:

$$\begin{gathered} x^2-6x+2x-12<0 \\ \Rightarrow x\left(x-6\right)+2\left(x-6\right)<0 \\ \Rightarrow \left(x+2\right)\left(x-6\right)<0 \end{gathered}$$

So the solution of the inequality is $-2<x<6$.

So if the function $f\left(x\right)$ is decreasing function, then $x\in \left(-2,6\right)$.

Hence, the correct option is C.