Find the values of x, such that f(x) is increasing function: f(x)… - Vidyadip

Question

Find the values of $x$, such that $f(x)$ is increasing function:
$f(x)=2 x^3-15 x^2-144 x-7$

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Answer

$
\begin{aligned}
& f(x)=2 x^3-15 x^2-144 x-7 \\
& \therefore f^{\prime}(x)=\frac{d}{d x}\left(2 x^3-15 x^2-144 x-7\right) \\
& =2 \times 3 x^2-15 \times 2 x-144 \times 1-0 \\
& =6 x^2-30 x-144 \\
& =6\left(x^2-5 x-24\right)
\end{aligned}
$
$f$ is increasing if, $f^{\prime}(x)>0$
i.e. if $6\left(x^2-5 x-24\right)>0$
i.e. if $x^2-5 x-24>0$
i.e. if $x^2-5 x>24$
i.e. if $x^2-5 x+\frac{25}{4}>24+\frac{25}{4}$
i.e. if $\left(x-\frac{5}{2}\right)^2>\frac{121}{4}$
i.e. if $x-\frac{5}{2}>\frac{11}{2}$ or $x-\frac{5}{2}<-\frac{11}{2}$
i.e. if $x >8$ or $x <-3$
i.e. if $x \in(-\infty,-3) \cup(8, \infty)$
$\therefore f$ is increasing, if $x \in(-\infty,-3) \cup(8, \infty)$.

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