CBSE 2025 · Region 1 · Set 1 · Q24 · 2 marks
Find the intervals in which function $\displaystyle \mathrm{f}(x)=5 x^{\frac{3}{2}}-3 x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.
Marking-scheme solution
\[\mathrm{f}(x)=5 x^{3 / 2}-3 x^{5 / 2} \Rightarrow \mathrm{f}^{\prime}(x)=\frac{15}{2} \sqrt{x}(1-x)
\]
For increasing / decreasing,put $\displaystyle \boldsymbol{\mathrm{f}}^{\prime}(\boldsymbol{x})=\mathbf{0}$
\[\Rightarrow x=0,1
\]
(i)
When $\displaystyle x \in[0,1], \mathrm{f}^{\prime}(x) \geq 0$. So, $\displaystyle \mathrm{f}$ is increasing when $\displaystyle x \in[0,1]$
(The intervals( $\displaystyle 0,1 ),[0,1) \operatorname{or}(0,1]$ can alsobeconsidered.)
(ii)
When $\displaystyle x \in[1, \infty), \mathrm{f}^{\prime}(x) \leq 0$.So, $\displaystyle \mathrm{f}$ is decreasing when $\displaystyle x \in[1, \infty)$
(The interval( $\displaystyle 1, \infty$ )can also beconsidered.)
Application of DerivativesIncreasing and Decreasing FunctionsApplyvery_short_answermedium
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CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.