Advertisements
Advertisements
Question
The function f(x) = cot−1 x + x increases in the interval
Options
(1, ∞)
(−1, ∞)
(−∞, ∞)
(0, ∞)
Advertisements
Solution
(−∞, ∞)
\[f\left( x \right) = \cot^{- 1} x + x\]
\[f'\left( x \right) = \frac{- 1}{1 + x^2} + 1\]
\[ = \frac{- 1 + 1 + x^2}{1 + x^2}\]
\[ = \frac{x^2}{1 + x^2} \geq 0, \forall x \in R\]
\[\text { So,f(x)is increasing on } \left( - \infty , \infty \right) .\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
For every value of x, the function f(x) = `1/7^x` is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
If f(x) = x + cosx – a then ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
