Which fraction exceeds its pth power by the greatest number possible?

If the sum of the hypotenuse and a side of a right-angled triangle is given. Show that the area of the triangle is maximum when the angle between them is

OR

A manufacturer can sell x items at a price of Rs. each. The cost price of x items is Rs. Find the number of items he should sell to earn maximum profit.

A metal box with a square base and vertical sides is to contain 1024 cm³. The material for the top and bottom costs ` 5 per cm<sup>2</sup> and the material for the sides costs ` 2.50 per cm². Find the least cost of the box.

Show that among all positive numbers x and y with x² + y² = r², the sum x + y is largest when x = y = .

Find the local maxima and local minima, of the function f(x) sin x – cos x, 0< x < 2π .Also, find the local maximum and local minimum values.

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is cot-1√2.

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

OR

An open box with a square base is to be made out of a given quantity of cardboard of area c² square units. Show that the maximum volume of the box iscubic units.

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is