The function f(x)=^(-1)x+x increases in the interval: Option 1: (1,∞) Option 2: (-∞,0) Option 3: (-∞,∞) Option 4: (0,∞)

Answer: (-∞,∞). Solution: f'(x)=1-1/(1+x^2)=x^2/(1+x^2)≥0 for every real x, so f increases on the whole real line.

Calculus

Application of Derivatives

Easy

core

The function $f(x)=\cot^{-1}x+x$ increases in the interval:

$(1,\infty)$

$(-\infty,0)$

$(-\infty,\infty)$

$(0,\infty)$

✅ Correct Option: 3

$f'(x)=1-\frac{1}{1+x^2}=\frac{x^2}{1+x^2}\ge0$ for every real $x$ , so $f$ increases on the whole real line.

2026: 22 May Shift 2

The function $f(x)=x^2-x\sqrt{2}$ is strictly decreasing in the interval :

2022: 17 Aug Shift 2

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$ . The value of $k$ is :

2023: 22 May Shift 3

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

2025: 16 May Shift 1

If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$ , x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

2024: 16 May Shift 1

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$ , $x \in [0, 3]$ , match List-I with List-II :

2022: 17 Aug Shift 2

Based on above information answer the following question :

If area of the flower bed is maximum, then area (in $m^2$ ) of the garden, which is outside the flower bed is :