Advertisements
Advertisements
प्रश्न
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
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) = 10 − 6x − 2x2 ?
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) = 5x3 − 15x2 − 120x + 3 ?
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) = 2x3 − 9x2 + 12x − 5 ?
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 \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
The function f(x) = cot−1 x + x increases in the interval
The function f(x) = xx decreases on the interval
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
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.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = x2 – 2x is increasing in the interval ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
