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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions5 Marks
Question
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$
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Answer

$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$
$\text{f}'(\text{x})=\frac{15}{2}\text{x}^{\frac{1}{2}}-\frac{15}{2}\text{x}^{\frac{3}{2}}$
$=\frac{15}{2}\text{x}^{\frac{1}{2}}(1-\text{x})$
Here, 0, 1 are the roots.
The possible intervals are $(-\infty,0),(0,1),(1,2)$ and $(1,\infty)\ ....(1)$
For f(x) to be increasing, we must have
$\text{f}'(\text{x})>0$
$\Rightarrow\frac{15}{2}\text{x}^{\frac{1}{2}}(1-\text{x})<0$
$\Rightarrow\text{x}\in(0,1)$
So, f(x) is increasing on (0, 1).
For f(x) to be decreasing, we must have,
$\text{f}'(\text{x})<0$
$\Rightarrow\frac{15}{2}\text{x}^{\frac{1}{2}}(1-\text{x})<0$
$\Rightarrow\text{x}\in(1,\infty)$
So, f(x) is decreasing on $\text{x}\in(1,\infty).$

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