The function f(x) = (x - 2)/(x + 1), x ≠ -1 is increasing when (Where R is a set of real numbers) Option 1: x R Option 2: x R - \-1\ Option 3: x R - \1, -1\ Option 4: x R - \0\

Answer: x R - \-1\. Solution: A function is increasing when its derivative f'(x) > 0. Given f(x) = (x - 2)/(x + 1) Using the quotient rule with u = x - 2 and v = x + 1: f'(x) = ((x+1)(1) - (x-2)(1))/((x+1)^2) f'(x) = (x + 1 - x + 2)/((x+1)^2) f'(x) = 3/((x+1)^2) For f'(x) > 0: 3/((x+1)^2) > 0 The numerator 3 is always positive. The denominator (x+1)^2 is always positive for all x ≠ -1. Therefore f'(x) > 0 for all x ≠ -1. The function is increasing for x R - \-1\. This represents all real numbers except -1.

Calculus

Application of Derivatives

Medium

core

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

$x \in \mathbb{R}$

$x \in \mathbb{R} - \{-1\}$

$x \in \mathbb{R} - \{1, -1\}$

$x \in \mathbb{R} - \{0\}$

✅ Correct Option: 2

A function is increasing when its derivative $f'(x) > 0$ .

Given $f(x) = \frac{x - 2}{x + 1}$

Using the quotient rule with $u = x - 2$ and $v = x + 1$ :

$f'(x) = \frac{(x+1)(1) - (x-2)(1)}{(x+1)^2}$

$f'(x) = \frac{x + 1 - x + 2}{(x+1)^2}$

$f'(x) = \frac{3}{(x+1)^2}$

For $f'(x) > 0$ :

$\frac{3}{(x+1)^2} > 0$

The numerator $3$ is always positive.

The denominator $(x+1)^2$ is always positive for all $x \neq -1$ .

Therefore $f'(x) > 0$ for all $x \neq -1$ .

The function is increasing for $x \in \mathbb{R} - \{-1\}$ .

This represents all real numbers except $-1$ .

2026: 30 May Shift 2

If $f(x) = (x+3)e^{-x}$ , then which of the following statements are True?

(A) Function $f(x)$ is increasing on $(-2, \infty)$ .

(B) Function $f(x)$ is decreasing on $(-2, \infty)$ .

(D) Function $f(x)$ is decreasing on $(2, 3)$ .

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The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

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The maximum value of $x^{-x}$ is

2026: 22 May Shift 1

The function $f:R\to R$ is defined by $f(x)=2x^3+5$ , then which of the following statements are correct ?

(A) $f(x)$ has no local maximum value.

(B) $f(x)$ has no local minimum value.

(D) $f(x)$ has neither a local maximum value nor a local minimum value

Choose the correct answer from the options given below:

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The demand function for a certain product is represented by $p = 35 + 5x - 3x^2$ , where $x$ is the number of units demanded and $p$ is the price per unit. The marginal revenue when 2 units are sold is

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The function $f(x) = x^2 - 4x + 6$ is

(A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$

(B) Strictly increasing on $(2, \infty)$

(D) Strictly decreasing on $(-\infty, 2)$

Choose the correct answer from the options given below:

The function f(x)=x−2x+1,x≠−1f(x) = \frac{x - 2}{x + 1}, x \neq -1f(x)=x+1x−2​,x=−1 is increasing when (Where R\mathbb{R}R is a set of real numbers)

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CUET Mathematics Past Year Questions (Topic-Wise)

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)