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MaMaxxiimmaa MMiinniimmaa
Some Important results :
Sphere of radius ‘r’:
Volume r 3
2 Surface 4 r
Right circular cylinder of height ‘h’ and radius of the base ‘r’:
2 Volume r h curved surface 2 rh
2 Total surface 2 rh 2 r
Right circular cone of height ‘h’ and radius of the base ‘r’:
(^12) Volume r h 3
curved surface rL
Where ‘L’ is the slant height such that (^)
2 2 L r h
- Find all the points of maxima and minima and the corresponding values:-
(a)
4 3 2 f (x) x 2x 3x 4x 4 (h)
3 2 f (x) 2x 9x 12x 1.
(b)
3 2 f (x) x 12x 36x 21. (i) Sin2x in [0^ ]
(c)
3 2
f (x) x x 16x 16. (j) Sin2x x in [ ]
(d)
f (x) x x 6x 8 3 2
(k) Sinx Cos2x in [0 ]
^
(e) Sinx Cos2x in [0 2 ] (l)
x f (x) in [1 4] (x 1)(x 4)
(f) Sinx in [0 2 ] (m)
f (x) x 8x x 105 4 2
(g) Cosx in [0 2 ] (n) f (x) x 1 x where x 0
- Prove that
x f (x) 1 x(Tanx)
is maximum when x Cosx.
- Show that the maximum value of
x 1
x
is
1 e
e.
- Show that (^)
p (^) q Sin Cos attains maximum when
1 p Tan q
.
- Show that the maximum value of
1
x xis
1 e
e.
- Show that the maximum value of
log x
x
is
2e
.
- Divide ‘a’ into two parts such that the product of the pth power of one and qth power of the other is as great
as possible.
- Show that the maximum rectangle inscribed in a circle is a square.
- Show that of all the rectangles with a given perimeter, the square has the largest area.
- Show that of all the rectangles with a given area, the square has the smallest perimeter.
MaMaxxiimmaa MMiinniimmaa
- The sum of one number and three times a second number is 60. Among the possible numbers satisfying this
condition, find the pair whose product is maximum.
- If the sum of the lengths of the hypotenuse and another side of a right angled triangle is given, show that
the area of the triangle is a maximum when the angle between these sides is
- An open rectangular tank with a square base and vertical sides is to be constructed of sheet metal to hold a
given quantity of water. Show that the cost of material will be least when the depth is half the width.
- A figure consists of a semi-circle with a rectangle on its diameter. Given that the perimeter of the figure is
20 meters, find its dimensions in order that its area is maximum.
- A piece of wire of length ‘l’ is cut into two parts, one of which is bent in the shape of a circle and the other
into the shape of a square. How the wire should be cut so that the sum of the areas of the circle and the
square is minimum.
- The three sides of a trapezium are equal, each being 6cms long; find the area of the trapezium when it is
maximum.
17. Show that the semi-vertical angle of the cone of maximum volume and of given slant height is
1 Tan 2
- Prove that 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 semi-vertical angle of the right circular cone of given total surface (including area of the
base) and maximum volume is
Sin 3
^
.
- The sum of the perimeter of a circle and a square is given. Show that when the sum of the areas is least the
side of the square is equal to the diameter of the circle.
- Show that the volume of the greatest cylinder which can be inscribed in a cone of height ‘h’ and semi-
vertical angle ‘A’ is
h Tan A 27
- Show that the height of cylinder of max. volume that can be inscribed in a sphere of radius ‘a’ is 2a^ 3
- Prove that the area of a right angled triangle of a given hypotenuse is maximum when the triangle is
isosceles.
- A square piece of tin of side 18cm is to be made into a box without the top by cutting a square piece from
each corner 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? Also, find the maximum volume of the box.
