Advertisements
Advertisements
प्रश्न
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Advertisements
उत्तर
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.
APPEARS IN
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
The function f(x) = cot−1 x + x increases in the interval
Every invertible function is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
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.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
For every value of x, the function f(x) = `1/7^x` is ______
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
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.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
