The total cost of manufacturing x articles is C = 47x + 300x2 - x4. Find x, for which average cost is increasing - Mathematics and Statistics

Advertisements
Advertisements
Sum

The total cost of manufacturing x articles is C = 47x + 300x2 − x4.  Find x, for which average cost is increasing.

Advertisements

Solution

Given,
Total cost function is (C) = 47x + 300x2 – x4 
Average cost CA = `"C"/"A"`

∴ CA = `(47 x + 300x^2 – x^4)/x`

∴ CA = `(x(47 + 300x – x^3))/x`

∴ CA = 47 + 300x – x3

`"dC"_"A"/"dx" = "d"/"dx" 47 + 300x  –  x^3`

∴ `"dC"_"A"/"dx"` = 0 + 300 – 3x2

∴ `"dC"_"A"/"dx"` = 3(100 –  x2)

Since average cost, CA is an increasing function, `"dC"_"A"/"dx" > 0`

∴ 3(100 – x2) > 0

∴ 100 – x2 > 0

∴ 100 > x2

∴ x2 < 100

∴ – 10 < x < 10

∴ x > – 10 and x < 10

But x > – 10 is not possible.     ...[∵ x > 0]

∴ x < 10  

∴ The average cost CA is increasing for x < 10. 

  Is there an error in this question or solution?
Chapter 4: Applications of Derivatives - Exercise 4.4 [Page 112]

RELATED QUESTIONS

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


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

Find the intervals in which the following functions are strictly increasing or decreasing:

 (x + 1)3 (x − 3)3


The interval in which y = x2 e–x is increasing is ______.


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) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?


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) = 8 + 36x + 3x2 − 2x?


Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?


Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?


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(x) =  \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\]  x > 0 ?


Show that f(x) = e2x is increasing on R.


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 f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?


Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?


Find the intervals in which f(x) = log (1 + x) −\[\frac{x}{1 + x}\] is increasing or decreasing ?


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).


Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?


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) = x|x|, x \[\in\] R ?


Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?


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


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


Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?


State whether f(x) = tan x − x is increasing or decreasing its domain ?


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


If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then


If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, 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.


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


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


Using truth table show that ∼ (p → ∼ q) ≡ p ∧ q 


 Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R. 


Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.


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) = `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 y = xx, (x > 0) is increasing and decreasing.


Solve the following:

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


Test whether the following function is increasing or decreasing.

f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0


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

f(x) = 2x3 - 15x2 + 36x + 1 


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

f(x) = 2x3 - 15x2 - 144x - 7 


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.


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 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 f(x) = 2x3 – 15x2 – 144x – 7 is

(a) Strictly increasing
(b) strictly 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`


A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong 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


For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.


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 ______ 


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 ______


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


For every value of x, the function f(x) = `1/7^x` is ______ 


If f(x) = x3 – 15x2 + 84x – 17, then ______.


The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.


Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.


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


The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.


In `(0, pi/2),`  the function f (x) = `"x"/"sin x"` is ____________.


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).


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


Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.


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


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


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


Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.


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))` = ______.


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 ______.


The function f(x) = tan–1(sin x + cos x) is an increasing function 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 function f(x) = sin4x + cos4x is an increasing function if ______.


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×