Advertisements
Advertisements
Question
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Advertisements
Solution
y = `log (1 + x) – (2x)/(2 + x), x > - 1`
∴ `dy/dx =d/dx[log (1 + x) - (2x)/(2 + x)]`
= `(1)/(1 + x).d/dx(1 + x) - ((2 + x).d/dx(2x) - 2x.d/dx(2 + x))/(2 + x)^2`
= `(1)/(1 + x) xx (0 + 1) ((2 + x) xx 2 - 2x(0 + 1))/(2 + x)^2`
= `(1)/(1 + x) - (4 + 2x - 2x)/(2 + x)^2`
= `(1)/(1 + x) - (4)/(2 + x)^2`
= `((2 + x)^2 - 4(1 + x))/((1 + x)(2 + x)^2)`
= `(4 + 4x + x^2 - 4 - 4x)/((1 + x)(2 + x)^2`
= `(x^2)/((1 + x)(2 + x)^2) >` 0 for all x > – 1
Hence, the given function is increasing function on its domain.
APPEARS IN
RELATED QUESTIONS
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) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
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) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
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) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
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.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
The function f(x) = sin x + 2x is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
