Determine the values of x for which f(x) = (x - 3)/(x + 1), x ≠ -1 is an increasing function.

f is increasing for all xR-\-1\

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2026: question 23

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerApplication of Derivatives

Determine the values of x for which $f (x) = \frac{x - 3}{x + 1}$ , $x \neq = - 1$ is an increasing function.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

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

Hint 1

A function is increasing wherever $f^{'} (x) > 0$ .

Hint 2

Use the quotient rule and simplify the numerator.

Hint 3

Check the sign of $(x + 1)^{2}$ for $x \neq = - 1$ .

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.

By the quotient rule, $f^{'} (x) = \frac{( x + 1 ) \cdot 1 - ( x - 3 ) \cdot 1}{( x + 1 ) ^{2}}$
1 mark
$f^{'} (x) = \frac{4}{( x + 1 ) ^{2}} > 0$ for every $x \neq = - 1$ , since $(x + 1)^{2} > 0$ .
½ mark
Hence $f$ is increasing for all $x \in R - {- 1}$ , i.e. on $(- \infty, - 1)$ and on $(- 1, \infty)$ .
½ mark

Answer: $f$ is increasing for all $x \in R - {- 1}$

Where the marks are

Correct quotient-rule derivative (1 mark)
Simplifies to $\frac{4}{( x + 1 ) ^{2}} > 0$ (½ mark)
Concludes increasing on $R - {- 1}$ (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 paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 23

Hints

Step-by-step solution

Revise the chapter

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