Show that the maximum and minimum values of  where  are given by the roots of the equation (

Two ships are sailing with velocities u and v towards the origin along courses which are inclined at an angle θ to each other. If at certain instant they are at distance a and b from the origin, find the minimum distance apart.

A given quantity of metal is to be cast into a half–cylinder i.e. with rectangular base and semicircular ends. Show that, in order for the total surface area to be minimum, the ratio of the length of the cylinder to the diameter of its semicircular ends is .

A poster is to contain 1600 square centimeters of printed matter with margins of 4 centimeters each at the top and the bottom and 2 centimeters at each side. Find the overall dimensions if the total area of the poster is minimum.

A lane runs at right angles to a road a feet wide. Find how many feet wide the lane must be if it just possible to carry a pole b feet long (b > a) from the road into the lane keeping it horizontal?

Show that the volume of the greatest cylinder which can be inserted in a cone of height h and semi vertical angle α is .

A wire three meters long has to be bent into the form of a rectangle with external loop at one corner and the rectangle is to have one side double the other. Find the dimensions of the rectangle and the radius of the circle so that the total enclosed area is a minimum.

From a circular sheet of metal it is required to cut a sector so that the remainder may be bent into a cone shaped vessel of maximum capacity. Find the angle of the sector to be removed.

How close does the curve  come to Y– axis?

Investigate whether it is possible to find numerical values of a and b such that y =  shall have a minimum value equal to 1 when .

Show that the rectangle of maximum area which can be inscribed in a given circle is a square.

Find the area of the greatest isosceles triangle which can be inscribed in a given ellipse, the triangle having the vertex coincident with one extremity of the major axis.

Find the dimensions and volume of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius a.

Find the point on the ellipse  the tangent at which encloses with the axes a triangle of minimum area.

One corner of a rectangular sheet of paper of width one foot is folded so as to reach the opposite edge of the sheet. Find the minimum length of the crease.

Prove that a conical tent of given capacity will require least amount of canvas when its height is  times the radius of the base.

Find the right circular cone of minimum volume that can be circumscribed about a sphere of given radius, r.

Assuming that the petrol burnt in driving a motor boat varies as the cube of the velocity, show that the most economical speed when going against a current c miles per hour is  miles per hour.

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 the metal will be least when depth is half of width.

Show that the semi-vertical angle of a cone of given total surface and maximum volume is .

Find the minimum length of intercept made by the axes on the tangent to the ellipse

Find the minimum of the sum of the intercepts made by the tangent to the ellipse  on the axes.

Tangents are drawn to the ellipse  and the circle  at the points where a common ordinate cuts them. Show that if θ be the greatest angle included between the tangents then  .