Advertisements
Advertisements
प्रश्न
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
Advertisements
उत्तर
Given: Number of units = x
Selling price of each unit = ₹ (180 − x)
∴ Selling price of x unit = ₹ (180 − x) . x
= ₹ (180x − x2)
Also, cost price of x units = ₹ (x2 + 60x + 50)
Now, Profit = P = Selling price – Cost price
= (180x − x2) − (x2 + 60x + 50)
= 180x − x2 − x2 − 60x − 50
∴ P = −2x2 + 120x − 50
∴ `(dP)/dx = −4x + 120`
and `(d^2P)/dx^2 = -4`
∴ For maxima and minima,
Consider, `(dP)/dx = 0`
∴ 4x + 120 = 0
∴ −4x = −120
∴ x = 30
For x = 30,
`("d"^2"P")/"dx"^2` = −4 < 0
∴ Total profit is maximum when the number of units = 30.
संबंधित प्रश्न
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
Solve the following : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The function y = 1 + sin x is maximum, when x = ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of sin x . cos x is ______.
The maximum value of `(1/x)^x` is ______.
The function `"f"("x") = "x" + 4/"x"` has ____________.
Divide 20 into two ports, so that their product is maximum.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
Find the maximum and the minimum values of the function f(x) = x2ex.
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
