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Find the maximum and the minimum values, if any, without using derivatives of the following functions:
f(x) = 4x2 – 4x + 4 on R
f(x) = –(x – 1)2 + 2 on R
f(x) = |x + 2| on R
f(x) = sin 2x + 5 on R
f(x) = |sin 4x + 3| on R
f(x) = 2x3 + 5 on R
f(x) = – |x + 1| + 3 on R
f(x) = 16x2 –16x + 28 on R
f(x) = x3 – 1 on R
Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:
f(x) = (x – 5)4
f(x) = x3 – 3x
f(x) = x3 (x – 1)2
f(x) = (x – 1) (x + 2)2
f(x) = x3 – 6x2 + 9x +15
f(x) = sin 2x, 0 < x < π
f(x) = sin x – cos x, 0 < x < 2π
f(x) = cos x, 0 < x < π
f(x) = x3(2x – 1)3
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of each of the following functions. Also, find the points of inflection, if any:
f(x) = x4 – 62x2 + 120x + 9
f(x) = x3 – 6x2 + 9x + 15
f(x) = 2/x – 1/x2, x > 0
f(x) = x ex
f(x) = x/2 + 2/x, x > 0
f(x) = (x + 1) (x + 2)1/3, x ≥ –2
f(x) = x3 – 2ax2 + a2 x, a > 0, x ∈ R
Find the local extremum values of the following functions:
f(x) = (x – 1) (x – 2)2
f(x) = – (x – 1)3(x + 1)2
The function y = a log x + bx2 + x has extreme values at x = 1 and x = 2. Find a and b.
Show that has a maximum value at x = e
Find the maximum and minimum values of the function .
Find the maximum and minimum values of f(x) = tan x – 2x.
If f(x) = x3 + ax2 + bx + c has a maximum at x = – 1 and minimum at x = 3. Determine a, b and c.
Prove that f(x) = sin x + √3 cos x has maximum value at x = π/6.
Find the absolute maximum and the absolute minimum values of the following functions in the given intervals:
f(x) = 4x – x2/2 in [–2, 45]
f(x) = (x – 1)2 + 3 in [–3, 1]
f(x) = 3x4 – 8x3 + 12x2 – 48x + 25 in [0, 3]
in [1, 9]
Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].
Find the absolute maximum and minimum values of the function f given by f(x) = cos2x + sin x, x ∈ [0, π].
Find absolute maximum and minimum values of a function f given by f(x) = 12x4/3 – 6x1/3, x ∈ [–1, 1].
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 – 15x2 + 36x + 1 on the interval [1, 5].
Determine two positive numbers whose sum is 15 and the sum of whose squares is minimum.
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
How should we choose two numbers, each greater than or equal to –2, whose sum is 1/2 so that the sum of the first and the cube of the second is minimum?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
A beam is supported at the two ends and is uniformly loaded. The bending moment M at a distance x from one end is given by
Find the point at which M is maximum in each case.
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 lengths of the two pieces so that the combined area of the circle and the square is minimum?
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the wire should be cut so that the sum of the areas of the square and triangle is minimum?
Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when one side of the square is equal to diameter of the circle.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
Two sides of a triangle have lengths ‘a’ and ‘b’ and the angle between them is θ. What value of θ will maximize the area of the triangle? Find the maximum area of the triangle also.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find this maximum volume
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost Rs 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimensions of the rectangular part of the window to admit maximum light through the whole opening.
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle that will produce the largest area of the window.
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is .
A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find also the area.
Prove that a conical tent of given capacity will require the least amount of canvas when the height is √2 times the radius of the base.
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 2/3 of the diameter of the sphere.
Prove that the semi - vertical angle of the right circular cone of given volume and least curved surface is .
An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of the triangle is maximum when .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6√3r.
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.
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum?
Show what the maximum volume of the cylinder which can be inscribed in a sphere of radius cm is 500 π cm3 .
Show that among all positive numbers x and y with x2 + y2 = r2, the sum x + y is largest when x = y = .
Determine the points on the curve x2 = 4y which are nearest to the point (0, 5).
Find the point on the curve y2 = 4x which is nearest to the point (2, –8).
Find the point on the curve x2 = 8y which is nearest to the point (2, 4).
Find the point on the parabolas x2 = 2y which is closest to the point (0, 5).
Find the coordinates of a point on the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x – 3.
Find the point on the curve y2 = 2x which is at a minimum distance from the point (1, 4).
Find the maximum slope of the curve y = –x3 + 3x2 + 2x – 27
The total cost of producing x radio sets per days is Rs and the price per set at which they may be solid is Rs . Find the daily output to maximize the total profit.
Manufactures can sell x items at a price of Rs each. The cost price is Rs . Find the number of items he should sell to earn maximum profit.
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead will be least, if depth is made half of width.
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semi - circular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi - circular ends is π : (π + 2).
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a.
A straight line is drawn through a given point P (1, 4). Determine the least value of the sum of the intercepts on the coordinate axes.
The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
The space s described in time t by a particle moving in a straight line is given by s = t5 – 40t3 + 30t2 + 80t – 250 Find the minimum value of acceleration.
A particle is moving in a straight line such that its distance s at any time t is given by . Find when its velocity is maximum and acceleration minimum.
#Mark the correct alternative in each of the following
The maximum value of x1/x, x > 0 is
If for all positive x where a, b, > 0, then
The minimum value of is
For the function
Let f(x) = x3 + 3x2 – 9x + 2. Then f(x) has
The minimum value of f(x) = x4 – x2 – 2x + 6 is
The number which exceeds its square by the greatest possible quantity is
Let f(x) = (x – a)2 + (x – b)2 + (x – c)2. Then, f(x) has a minimum at x =
The sum of two non-zero numbers is 8, the minimum value of the sum of their reciprocals is
The function assumes minimum value at x =
At f(x) = 2sin 3x + 3 cos 3x is
If x lies in the interval [0, 1], then the least value of x2 + x + 1 is
The least value of the function f(x) = x3 – 18x2 + 96x in the interval [0, 9] is
The least value of the function on [–1, 1] is
The point on the curve y2 = 4x which is nearest to the point (2, 1) is
If x + y = 8, then the maximum value of xy is
The least and greatest values of f(x) = x3 – 6x2 + 9x in [0, 6], are
is maximum when x =
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the he the cone to the diameter of the sphere is
The minimum value of is
If , then its greatest value is
If then its maximum value is
Let x, y be two variables and x > 0, xy = 1, then minimum value of x + y is
f(x) = 1 + 2 sinx + 3cos2 x, 0 ≤ x ≤ is
The function f(x) = 2x3 – 15x2 + 36x + 4 is maximum at x =
The maximum value of on [–1, 1] is
Let f(x) = 2x3 – 3x2 – 12x + 5 on [–2, 4]. The relative maximum occurs at x =
The minimum value of x loge x is equal to
The minimum value of the function f(x) = 2x3 – 21x2 + 36x – 20 is
Write necessary condition for a point = c to be an extreme point of the function ().
Write sufficient conditions for a point = c to be a point of local maximum.
If f() attains a local minimum at = c, then write the values of ’(c) and ’’ (c).
Write the minimum value of .
Write the maximum value of
Write the point where () = e attains minimum value.
Find the least value of where a>0, b>0 and >0.
Write the minimum value of () = x.
Write the minimum value of () = 1/x.
Write the maximum value of if it exists.