In which of the following interval, the function f(x) = x/( x) is decreasing? Option 1: (-∞, e) Option 2: (0, e) Option 3: (0, e) - \1\ Option 4: (e, ∞)

Answer: (0, e) - \1\. Solution: The function f(x) = x/( x) is decreasing when its derivative is negative. The domain requires x > 0 and x ≠ 0, so x ≠ 1. Domain: x (0, ∞) - \1\ Using the quotient rule with u = x and v = x: f'(x) = ((1) * ( x) - (x) * (1/x))/(( x)^2) f'(x) = ( x - 1)/(( x)^2) For the function to be decreasing, f'(x) Combining with the domain: x 0 and x ≠ 1 Therefore, the function is decreasing on (0, e) - \1\.

Calculus

Application of Derivatives

Medium

applied

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?

$(-\infty, e)$

$(0, e)$

$(0, e) - \{1\}$

$(e, \infty)$

✅ Correct Option: 3

The function $f(x) = \frac{x}{\log x}$ is decreasing when its derivative is negative.

The domain requires $x > 0$ and $\log x \neq 0$ , so $x \neq 1$ .

Domain: $x \in (0, \infty) - \{1\}$

Using the quotient rule with $u = x$ and $v = \log x$ :

$f'(x) = \frac{(1) \cdot (\log x) - (x) \cdot \left(\frac{1}{x}\right)}{(\log x)^2}$

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

For the function to be decreasing, $f'(x) < 0$ :

$\frac{\log x - 1}{(\log x)^2} < 0$

Since $(\log x)^2$ is always positive, the sign depends on the numerator:

$\log x - 1 < 0$

$\log x < 1$

$x < e$

Combining with the domain:

$x < e$ and $x > 0$ and $x \neq 1$

Therefore, the function is decreasing on $(0, e) - \{1\}$ .

2025: 22 May Shift 2

The rate of change of the area of a circle with respect to its radius $r$ , when $r = 3$ cm, is:

2025: 30 May Shift 2

The function, $f(x) = x - \frac{1}{x}$ is

2025: 27 May Shift 1

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

2026: 22 May Shift 2

For the function $f(x) = (x+1)^2(x-2)^2$ , which of the following statements are TRUE?

(A) $f(x)$ is decreasing on $(-\infty,-1)$

(B) $f(x)$ is increasing on $\left(-1,\frac{1}{2}\right)$

(D) $f(x)$ is increasing on $(2,\infty)$

Choose the correct answer from the options given below:

2025: 27 May Shift 1

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

2023: 15 June Shift 2

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

In which of the following interval, the function f(x)=xlog⁡xf(x) = \frac{x}{\log x}f(x)=logxx​ is decreasing?

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?