If x + y = 3 show that the maximum value of x2y is 4. - Mathematics and Statistics

Advertisements
Advertisements
Sum

If x + y = 3 show that the maximum value of x2y is 4.

Advertisements

Solution

x + y = 3

∴ y = 3 – x

Let T = x2y = x2(3 – x) = 3x2 – x3 

Differentiating w.r.t. x, we get

`"dT"/("d"x) = 6"x" - 3"x"^2`   ....(i)

Again, differentiating w.r.t. x, we get

`("d"^2"T")/("d"x^2) = 6 - 6"x"`    ...(ii)

Consider, `"dT"/("d"x) = 0`

∴ 6x – 3x2 = 0

∴ x = 2

For x = 2,

`(("d"^2"T")/"dx"^2)_(x = 2)` = 6 – 6(2)

= 6 – 12

= – 6 < 0

Thus, T, i.e., x2y  is maximum at x = 2

For x = 2, y = 3 – x = 3 – 2 = 1

∴ Maximum value of T = x2y = (2)2(1) = 4

  Is there an error in this question or solution?
Chapter 4: Applications of Derivatives - Miscellaneous Exercise 4 [Page 114]

RELATED QUESTIONS

Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x). 


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.


An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.


If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/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`. Also find maximum volume in terms of volume of the sphere


A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.


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 and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|


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:

`h(x) = sinx + cosx, 0 < x < pi/2`


What is the maximum value of the function sin x + cos x?


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.


Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.


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?


For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.


A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`


Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .


 A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle. 


 The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it. 


 Find the point on the straight line 2x+3y = 6,  which is closest to the origin. 


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. 


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. 


Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.


Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20


Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x


Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`


Find the maximum and minimum of the following functions : f(x) = x log x


Find the maximum and minimum of the following functions : f(x) = `logx/x`


Divide the number 30 into two parts such that their product is maximum.


Divide the number 20 into two parts such that sum of their squares is minimum.


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.


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.


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.


An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.


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?


A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?


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.


Show that among rectangles of given area, the square has least perimeter.


Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.


Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.


Choose the correct option from the given alternatives : 

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.


Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.


Solve the following : Show that of all rectangles inscribed in a given circle, the square has the maximum area.


Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.


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:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.


Solve the following : 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)`.


Solve the following: 

Find the maximum and minimum values of the function f(x) = cos2x + sinx.


Determine the maximum and minimum value of the following function.

f(x) = 2x3 – 21x2 + 36x – 20


Determine the maximum and minimum value of the following function.

f(x) = x log x


Determine the maximum and minimum value of the following function.

f(x) = `x^2 + 16/x`


Divide the number 20 into two parts such that their product is maximum.


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?


Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x


The function f(x) = x log x is minimum at x = ______.


Find the local maximum and local minimum value of  f(x) = x3 − 3x2 − 24x + 5


A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum


A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?


Divide the number 20 into two parts such that their product is maximum


A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.


A metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.

Solution: Let the dimensions of the rectangle be x cm and y cm.

∴ 2x + 2y = 36

Let f(x) be the area of rectangle in terms of x, then

f(x) = `square`

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme value, f'(x) = 0, we get

x = `square`

∴ f''`(square)` = – 2 < 0

∴ Area is maximum when x = `square`, y = `square`

∴ Dimensions of rectangle are `square`


By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima

Solution: f(x) = x3 – 9x2 + 24x

∴ f'(x) = `square`

∴ f''(x) = `square`

For extreme values, f'(x) = 0, we get

x = `square` or `square`

∴ f''`(square)` = – 6 < 0

∴ f(x) is maximum at x = 2.

∴ Maximum value = `square`

∴ f''`(square)` = 6 > 0

∴ f(x) is maximum at x = 4.

∴ Minimum value = `square`


If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.


Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______ 


If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.


The function y = 1 + sin x is maximum, when x = ______ 


The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.


Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.


Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.


Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`


Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.


Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.


The function `"f"("x") = "x" + 4/"x"` has ____________.


A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)


Range of projectile will be maximum when angle of projectile is


The function `f(x) = x^3 - 6x^2 + 9x + 25` has


The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is


For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`


The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is


The maximum value of the function f(x) = `logx/x` is ______.


Divide 20 into two ports, so that their product is maximum.


Read the following passage and answer the questions given below.

In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1.

  1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
  2. Find the critical point of the function.
  3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
    OR
    Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.

A function f(x) is maximum at x = a when f'(a) > 0.


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 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 ______.


If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, 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 function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.


Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.


A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.


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 ______.


Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.


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 ______.


The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.


The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.


The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.


A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, 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 ______.


Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.


Read the following passage:

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.

One complete of a four-cylinder four-stroke engine. The volume displace is marked
The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π cm2.

Based on the above information, answer the following questions:

  1. If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
  2. Find `(dV)/(dr)`. (1)
  3. (a) Find the radius of cylinder when its volume is maximum. (2)
    OR
    (b) For maximum volume, h > r. State true or false and justify. (2)

If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).


Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.


Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.


Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.

Solution: Let one part be x. Then the other part is 84 - x

Letf (x) = x2 (84 - x) = 84x2 - x3

∴ f'(x) = `square`

and f''(x) = `square`

For extreme values, f'(x) = 0

∴ x = `square  "or"    square`

f(x) attains maximum at x = `square`

Hence, the two parts of 84 are 56 and 28.


Find the maximum and the minimum values of the function f(x) = x2ex.


A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.


A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.


If x + y = 8, then the maximum value of x2y is ______.


Divide the number 100 into two parts so that the sum of their squares is minimum.


Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).


A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?


Determine the minimum value of the function.

f(x) = 2x3 – 21x2 + 36x – 20


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`


Share
Notifications



      Forgot password?
Use app×