Find the intervals in which function f(x)=5x³/2-3x^5/2 is (i) increasing (ii) decreasing.

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2025: question 24

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section B, question 24, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24 · 2 marks · Very short answerApplication of Derivatives

Find the intervals in which function $f (x) = 5 x^{\frac{3}{2}} - 3 x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Note the domain first: $x^{3/2}$ needs $x \geq 0$ .

Hint 2

Differentiate and take out the common factor $\frac{15}{2} x$ .

Hint 3

Check the sign of $1 - x$ on each side of $x = 1$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

The function is defined for $x \geq 0$ . Differentiate: $f^{'} (x) = \frac{15}{2} x^{1/2} - \frac{15}{2} x^{3/2} = \frac{15}{2} x (1 - x)$ . Setting $f^{'} (x) = 0$ gives $x = 0, 1$ .
1 mark
(i) For $x \in [0, 1]$ , $x \geq 0$ and $1 - x \geq 0$ , so $f^{'} (x) \geq 0$ . Hence $f$ is increasing on $[0, 1]$ .
½ mark
(ii) For $x \in [1, \infty)$ , $1 - x \leq 0$ , so $f^{'} (x) \leq 0$ . Hence $f$ is decreasing on $[1, \infty)$ .
½ mark

Answer: (i) Increasing on $[0, 1]$ ; (ii) decreasing on $[1, \infty)$

Where the marks are

Finds $f^{'} (x) = \frac{15}{2} x (1 - x)$ and critical points $x = 0, 1$ (1 mark)
States increasing on $[0, 1]$ ; open or half-open forms also accepted (½ mark)
States decreasing on $[1, \infty)$ ; $(1, \infty)$ also accepted (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Increasing and Decreasing Functions

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 24

Hints

Step-by-step solution

Revise the chapter

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