29) The maximum value of
28) For all real values of x, the minimum value of
(A) 0 (B) 1 (C) 3 (D) 1/3
27) The point on the curve which is nearest to the point (0, 5) is
26) Show that semi-vertical angle of the right circular cone of given surface area and
maximum volume is
25) Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is
24) Show that the right circular cone of least curved surface and given volume has an altitude equal to time the radius of the base.
23) Prove that the volume of the largest cone that can be inscribed in a sphere of radius r is 8/27 of the volume of the sphere.
22) 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?
21) Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
20. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
19) Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
18) A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square 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 ?
17. 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 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.
16. Find two positive numbers whose sum is 16 and the sum of whose cubes is
15) Find two positive numbers x and y such that their sum is 35 and the product
is a maximum.
14) Find two positive numbers x and y such that x + y = 60 and is maximum.
14. Find two numbers whose sum is 24 and whose product is as large as possible.
12. Find the maximum and minimum values of
11. It is given that at x = 1, the function attains its maximum value,
on the interval [0, 2]. Find the value of a.
10. Find the maximum value of in the interval [1, 3]. Find the
the maximum value of the same function in [–3, –1].