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(s) of x for which y = [x(x − 2)]2 is an increasing function.
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 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 + 9x2 + 12x + 20 ?
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) = x3 − 12x2 + 36x + 17 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Let f(x) = x3 − 6x2 + 15x + 3. Then,
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
Show that f(x) = x – cos x is increasing for all x.
The price P for the demand D is given as P = 183 + 120D − 3D2, then the value of D for which price is increasing, is ______.
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
