1.
Class XII
Application of derivative (Maxima and Minima)
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of
radius 𝑟 is
2.
4𝑟
3
,Also find the maximum volume in terms of volume of sphere.
Prove that the least perimeter of an isosceles triangle in which a circle of radius 𝑟 can be inscribed is
6 3𝑟.
3. If the sum of length of the hypoteneous and a side of an right angled triangle is given, show that the
𝜋
area of triangle is maximum, when angle between them is .
3
4.
Show that the semi-vertical angle of a cone of maximum volume and given slant height is cos −1
5.
Find the maximum and minimum value of 𝑓 𝑥 = 𝑠𝑒𝑐𝑥 + 𝑙𝑜𝑔𝑐𝑜𝑠 2 𝑥, 0 < 𝑥 < 2𝜋.
6.
Show that the volume of largest cone that can be inscribed in a sphere of radius 𝑟 is
8
27
1
3
.
of volume of the
sphere.
7. Show that the volume of a cylinder which can be inscribed in a cone of height h and semi-vertical angle
300 is
4
81
𝜋𝑕3 .
8. 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 the maximum volume.
9. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2
times radius of base.
10. A point on the hypoteneous of a right angled- triangle is at a distance 𝑎 and 𝑏 from the sides. Show that
2
3
the minimum length of the hypoteneous is 𝑎 + 𝑏
3
2 2
3
.
11. Tangent to the circle 𝑥 2 + 𝑦 2 = 4 at any point on it in the first quadrant make intercepts OA and OB
on 𝑥-axis and 𝑦-axis respectively. O being the centre of circle, find the minimum value of 𝑂𝐴 + 𝑂𝐵 .
12. A metal box with a square base and vertical sides is to contain 1024𝑐𝑚 3 .The material for the top and
bottom cost Rs. 5 per cm2 and the material for the sides costs Rs. 2.50 per cm2. Find the least cost of the
box.
13. Show that the semi-vertical angle of the right circular cone of the given total surface area and maximum
1
volume is sin−1 .
3
14. Find the maximum area of an isosceles triangle inscribed in the ellipse
one end of the major axis.
𝑥2
𝑦2
𝑎
𝑏2
+
2
= 1 , with its vertex at
15. An open box with square base is to be made out of the given quantity of sheet of area𝑎2 . So that the
maximum volume of the box is
𝑎3
.
6 3
16. Find the largest possible area of a right angle triangle whose hypoteneous is 5 cm long.
17. Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
18. If the length of three sides of a trapezium other than the base is 40 cm each, find the area of the
trapezium when it is maximum.
19. Prove that the semi-vertical angle of right circular cone of given volume and least curved surface area
is cot −1 2.