a) 0

b) 1

c) 5

d) 25

 
for more accurate result for c ε (a, b),
 
when
If  for every x ε (a, b) and a < c < b and (c, f(c)) is a point lying on the curve for which F(c) is maximum, then (c) is equal to

a)

b)

c)

d) 0

 
for more accurate result for c ε (a, b),
 
when
If is continuous everywhere and
 
for all a, then f(x) is a polynomial in x whose maximum possible degree is

a) 1

b) 2

c) 3

d) 4

Suppose f is defined on [a, b] and  and f. Show that c is not a minimum point.

Suppose f is defined on [a, b] and  and f. Show that c is neither a maximum nor a minimum point.

Let f(x) =   on [−2, 2]. Find the points of maximum and minimum of f(x).

A box is made out of a rectangular piece of cardboard by cutting out a square from each corner and folding up the four resulting flaps to make sides. Suppose the sides of the piece of cardboard are a and b. What size should the squares be if the box is to have maximum volume?

A wire of length L is cut into two pieces one of which is bent into a circle and the other into a square. How should the wire be cut to make the sum of the two enclosed area a maximum?

A ray of light travels from the point A with speed s₁ below the X–axis in the air and with a speed S₂ above the X–axis to the point B in glass. What is the quickest path from A to B?

Two points  and  are above the X – axis. Show that the shortest path from one point to the other via the X–axis is the one such that  in the figure below.

A right circular cylinder is cut from a log of wood in the form of a frustum of cone of height h and the radii of the end circular sections as a, b (a > b). What is the maximum volume of the cylinder?

a)

b)

c)

d)

Let
where  are all positive. Show that f(x) is strictly increasing on whole of the real line.

Let n be a positive integer and x be a positive number.

i. Show that

ii. Deduce from i) that when  then  and

Let a be a positive number, and set

i. Show that f is decreasing on the interval  and increasing on the interval  and hence show that for  we have

ii. From part i) deduce that  for every  and

 iii. For any  

A closed cylindrical can holding a given volume V has a radius r and height h. The top and the bottom of the can have to be cut out of the rectangular piece of the material. The material for the can cost a cents per unit area and the scrap can be sold for b cents per unit area b < a. How should r and h be chosen so as to minimize the cost of the materials?

From the shore a man observes a rotating beacon light. The light rotates once a minute and the spot of the light crosses a wall behind the man at the rate of 10 yards per second. The wall is perpendicular to the line from the man to the beacon. How far is the beacon from the wall?

A spherical balloon rests in a cone as in the figure. The balloon loses air and as it sinks, it slides down in the cone. Find the rate at which the centre of the balloon is descending, given the following data:
Radius of balloon: 2 feet
Height of centre of balloon above vertex of the cone: 5 feet
Rate of decrease of volume of the balloon: 1.5 cubic feet per second

A tractor lifts a load of hay by pulling a rope over a pulley. Find the rate at which the load of hay is rising: Given the following data:
Length of rope, tractor hitch to pulley: 15 yards
Height of pulley above tractor hitch: 9 yards
Speed of tractor: 30 yards per minute

A villain in a TV western is trying to break a telephone line by tying a rope to it and riding off on his horse. At what speed does the upper end of the rope move at the moment it becomes taut? Given the following data:
Length of rope: 50 feet
Height of the wire above saddle: 10 feet
Speed of the horse: 2500 feet per minute

A ladder 13 feet long leans against a high, vertical wall. The lower end, five feet from the wall is slipping away from the wall at 2 feet per second. How fast is the top of the ladder slipping down? If the man is on the ladder 8 feet above the ground, how fast is he approaching the ground?

A ladder 13 feet long leans against a wall 7 feet high. The ladder projects 3 feet over the top of the wall. The bottom of the ladder is slipping away from the wall. How fast is the bottom of the ladder slipping if the ladder slides along the top edge of the wall at two feet per second?

A wall 9 feet high is 21 feet 4 inches from a house. Find the length of the shortest ladder which will reach the house when the lower end is on the horizontal ground on the other side of the wall.

Prove that the maximum and minimum values of the function y =  are those values of y for which  is a perfect square.