The function f(x) = x-1x, x ∈ R, x ≠ 0 is increasing - Mathematics and Statistics

Advertisements
Advertisements
MCQ
True or False

The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing

Options

  • True

  • False

Advertisements

Solution

This Statement is True.

  Is there an error in this question or solution?
Chapter 1.4: Applications of Derivatives - Q.3

RELATED QUESTIONS

Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing


The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?


Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 


The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing


Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is

  1. Strictly increasing
  2. Strictly decreasing

Show that y = `log(1+x) - (2x)/(2+x), x> -  1`, is an increasing function of x throughout its domain.


Prove that  y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`


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 intervals in which the function f given by `f(x) = x^3 + 1/x^3` x != 0, is 

(i) increasing

(ii) decreasing

 


Let f be a function defined on [ab] such that f '(x) > 0, for all x ∈ (ab). Then prove that f is an increasing function on (ab).


Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.


Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?


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) = 10 − 6x − 2x2  ?


Find the interval in which the following function are increasing or decreasing  f(x) = x2 + 2x − 5  ?


Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x?


Find the interval in which the following function are increasing or decreasing  f(x) = 2x3 − 24x + 107  ?


Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?


Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?


Show that f(x) = x − sin x is increasing for all x ∈ R ?


Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?


Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?


Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?


State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?


Show that f(x) = tan−1 x − x is a decreasing function on R ?


Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?


Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?


Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?


Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?


What are the values of 'a' for which f(x) = ax is increasing on R ?


Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?


The interval of increase of the function f(x) = x − ex + tan (2π/7) is


Let f(x) = x3 − 6x2 + 15x + 3. Then,


Function f(x) = | x | − | x − 1 | is monotonically increasing when

 

 

 

 

 

 

 

 

 

 

 


In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is


The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if

 


If x = cos2 θ and y = cot θ then find `dy/dx  at  θ=pi/4` 


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.


Find `dy/dx,if e^x+e^y=e^(x-y)`


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).


 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.


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.


Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.


Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6


Find the values of x for which the following functions are strictly increasing : f(x) = 3 + 3x – 3x2 + x3 


Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7


Find the values of x for which the following functions are strictly decreasing:

f(x) = 2x3 – 3x2 – 12x + 6


Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`


Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12


Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing


Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.


show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.


Show that f(x) = x – cos x is increasing for all x.


Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.


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 ______.


Solve the following:

Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.


Find the value of x, such that f(x) is increasing function.

f(x) = x2 + 2x - 5 


Find the value of x, such that f(x) is decreasing function.

f(x) = 2x3 – 15x2 – 84x – 7 


Find the value of x such that f(x) is decreasing function.

f(x) = x4 − 2x3 + 1


For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.


Choose the correct alternative.

The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is


State whether the following statement is True or False:

The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.


Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.


Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______


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.


Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing


Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R


Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing


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 the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing


Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing


Choose the correct alternative:

The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is


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 ______.


The slope of tangent at any point (a, b) is also called as ______.


If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______


The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is  ______


State whether the following statement is True or False: 

The function f(x) = `3/x` + 10, x ≠ 0 is decreasing


State whether the following statement is True or False: 

If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1


Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function


Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function


Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing


By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.

Solution: f(x) = 2x3 – 15x2 – 84x – 7

∴ f'(x) = `square`

∴ f'(x) = 6`(square) (square)`

Since f(x) is decreasing function.

∴ f'(x) < 0

Case 1: `(square)` > 0 and (x + 2) < 0

∴ x ∈ `square`

Case 2: `(square)` < 0 and (x + 2) > 0

∴ x ∈ `square`

∴ f(x) is decreasing function if and only if x ∈ `square`


The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.


If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.


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 ______ 


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?


In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?


The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.


If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.


y = x(x – 3)2 decreases for the values of x given by : ______.


The function f(x) = tanx – x ______.


In case of decreasing functions, slope of tangent and hence derivative is ____________.


The function f(x) = tan-1 x is ____________.


The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.


The function f: N → N, where

f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is


Which of the following graph represent the strictly increasing function.


Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 –  h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.


Function given by f(x) = sin x is strictly increasing in.


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


Show that function f(x) = tan x is increasing in `(0, π/2)`.


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).


If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.


The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.


Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.


Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.


The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.


If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.


If f(x) = x + cosx – a then ______.


Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.


y = log x satisfies for x > 1, the inequality ______.


Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.


The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.


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 ______.


A function f is said to be increasing at a point c if ______.


Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.


Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.


The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.


Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.


Share
Notifications



      Forgot password?
Use app×