Question

The function $f\left(x\right)=x^3-3x^2-24x+5$ is an increasing function in the interval given below.

• A. $(-\infty ,–2)\cup (4,\infty )$
• B. $(-2,\infty )$
• C. $(-2,4)$
• D. $(-\infty ,4)$

Answer 1

The correct option is A

$(-\infty ,–2)\cup (4,\infty )$

Evaluating for the correct answer

$f\left(x\right)=x^3-3x^2-24x+5$

For, an increasing function $f\text{'}\left(x\right)>0$

$$\begin{gathered} \Rightarrow \frac{d\left(x^3-3x^2-24x+5\right)}{dx}>0 \\ \Rightarrow 3x^2-6x-24>0 \\ \Rightarrow x^2-2x-8>0 \\ \Rightarrow (x+2)(x-4)>0 \end{gathered}$$

Now, $f\text{'}\left(x\right)=(x+2)(x-4)>0$ in the interval

$x\in (-\infty ,-2)$ and $x\in (4,\infty )$

So, $f\left(x\right)$is an increasing function in the interval $x\in (-\infty ,-2)\cup (4,\infty )$

Hence, the correct option is option (A)