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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES2 Marks
Question
Find the intervals in which the following function are strictly increasing or decreasing:
$(x + 1)^3 (x - 3)^3$
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Answer

Given: $\text{f}\text{(x)} = (\text{x} + 1)^3(\text{x}-3)^3 $
$\Rightarrow\ \text{f}'\text{(x)} = (\text{x} + 1)^3.3(\text{x} - 3)^2 + (\text{x} -3)^3.3 (\text{x} + 1)^2 $
$\Rightarrow\ \text{f}'\text{(x)} = 3(\text{x}+1)^2(\text{x}- 3)^2(\text{x}+1+\text{x}-3)$
$\Rightarrow\ \text{f}'\text{(x)} =3(\text{x} + 1)^2 (\text{x}-3)^2(2\text{x}-2)$
$\Rightarrow\ \text{f}'\text{(x)} = 6(\text{x}+1)^2(\text{x}-3)^2(\text{x}-1)$
Here, factors $(\text{x} +1)^2 \text{ and } (\text{x}-3)^2$ are non-negative for all x.
Therefore, f(x) is strictly increasing if $\text{f}'\text{(x)} >0 \ \Rightarrow\ \text{x} - 1>0 \Rightarrow \text{x}> 1$
And f(x) is strictly decreasing if $\text{f}'\text{(x)} < 0\Rightarrow \ \text{x}-1< 0\ \Rightarrow \ \text{x} < 1$
Hence, f is strictly increasing in $(1,\ \infty)$ and f is strictly decreasing in $(-\infty,\ 1).$

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