Advertisements
Advertisements
Question
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Advertisements
Solution
Given that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)`
Differentiating both sides w.r.t. x, we get
f'(x) = `2 - 1/(1 + x^2) + 1/(sqrt(1 + x^2) - x) xx "d"/"dx" (sqrt(1 + x^2) - x)`
= `2 - 1/(1 + x^2) + ((1/(2sqrt(1 + x^2)) xx (2x - 1)))/(sqrt(1 + x^2) - x)`
= `2 - 1/(1 + x^2) + (x - sqrt(1 + x^2))/(sqrt(1 + x^2) (sqrt(1 + x^2 - x))`
= `2 - 1/(1 + x^2) - ((sqrt(1 + x^2) - x))/(sqrt(1 + x^2) (sqrt(1 + x^2) - x))`
= `2 - 1/(1 + x^2) - 1/sqrt(1 + x^2)`
For increasing function, f '(x) ≥ 0
∴ `2 - 1/(1 + x^2) - 1/sqrt(1 + x^2) ≥ 0`
⇒ `(2(1 + x^2) - 1 + sqrt(1 + x^2))/((1 + x^2)) ≥ 0`
⇒ `2 + 2x^2 - 1 + sqrt(1 + x^2) ≥ 0`
⇒ `2x^2 + 1 + sqrt(1 + x^2) ≥ 0`
⇒ `2x^2 + 1 ≥ - sqrt(1 + x^2)`
Squaring both sides, we get 4x4 + 1 + 4x2 ≥ 1 + x2
⇒ 4x4 + 4x2 – x2 ≥ 0
⇒ 4x4 + 3x2 ≥ 0
⇒ x2(4x2 + 3) ≥ 0
Which is true for any value of x ∈ R.
Hence, the given function is an increasing function over R.
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
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).
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 function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
The slope of tangent at any point (a, b) is also called as ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function `1/(1 + x^2)` is increasing in the interval ______
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Which of the following graph represent the strictly increasing function.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
