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 the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
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) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
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 ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and 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 f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
The function f (x) = x2, for all real x, is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Which of the following graph represent the strictly increasing function.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
