Find the interval for which function f(x)=x^3- 3 x^2-24 x+5 is in… - Vidyadip

Question

Find the interval for which function $f(x)=x^3-$ $3 x^2-24 x+5$ is increasing.

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Answer

Given $\quad$$f(x)=x^3-3 x^2-24 x+5$
or$\quad$$y=x^3-3 x^2-24 x+5$
$\Rightarrow \quad \frac{d y}{d x}=3 x^2-6 x-24$
$\Rightarrow \quad \frac{d y}{d x}=3\left(x^2-2 x-8\right)$$
\begin{array}{ll}
\Rightarrow & \frac{d y}{d x}=3(x-4)(x+2) \\
\text { For increasing } & \frac{d y}{d x}>0
\end{array}
$
So, $\quad 3(x-4)(x+2)>0$$
\Rightarrow \quad(x-4)(x+2)>0 \quad[\because 3>0]
$
Case I, $x-4>0,(x+2)>0 \Rightarrow x>4, x>-2$
Case II, $x-4<0,(x+2)<0 \Rightarrow x<4, x<-2$$
x \in(-\infty,-2) \cup(4, \infty)
$

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