Advertisements
Advertisements
प्रश्न
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Advertisements
उत्तर
Profit function P(x) is given by
P(x) = (150 – x)x – 1625
= 150x – x2 – 1625
∴ P'(x) = `"d"/("d"x)(150 x - x^2 - 1625)`
= 150 × 1 – 2x – 0
= 150 – 2x
and
P"(x) = `"d"/("d"x)(150 - 2x)`
= 0 – 2 × 1
= – 2
Now, P'(x) = 0 gives, 150 – 2x = 0
∴ x = 75
and
P''(75) = – 2 < 0
∴ by the second derivative test, P(x) is maximum when x = 75
Maximum profit = P(75)
= (150 – 75)75 – 1625
= 75 × 75 – 1625
= 4000
Hence, the profit will be maximum, if the manufacturer manufactures 75 items and maximum profit is 4000.
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3.`
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = x log x
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the 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 the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
If f(x) = x.log.x then its maximum value is ______.
Divide the number 20 into two parts such that their product is maximum
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The function y = 1 + sin x is maximum, when x = ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
The maximum value of `(1/x)^x` is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
If y = x3 + x2 + x + 1, then y ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The function `f(x) = x^3 - 6x^2 + 9x + 25` has
Divide 20 into two ports, so that their product is maximum.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
A maximum or minimum value of \[f\] is called a(n) _____.
What is an absolute minimum?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
