Mathematics Integration Level: Misc Level

The sum of two positive numbers is 20. Find the numbers if the sum of their squares is minimum

Mathematics Integration Level: Misc Level

The sum of two positive numbers is 20. Find the numbers if the product of the squares of one and the cube of the other is maixmum

Mathematics Integration Level: Misc Level

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is maximum

Mathematics Integration Level: Misc Level

Three numbers are given whose sum is 180 and the ratio of first two of them is 1:2 if the product of the numbers is greatest, find the numbers

Mathematics Integration Level: Misc Level

Prove that the perimeter of a right-angled triangle of given hypotenuse is maximum when the triangle is isosceles

Mathematics Integration Level: Misc Level

Prove that of all rectangles with given perimeter, the square has the largest area

Mathematics Integration Level: Misc Level

Two sides of at triangle have lengths a and b, and the angle between them is . What value of will maximize the area of triangle? Also find the maximum area of the triangle

Mathematics Integration Level: Misc Level

A wire 10 meters long is cut into two parts. One part is bent into the shape of a circle and the other into the shape of an equilateral triangle. How should the wire be cut so that the combined area of the two figures is as small as possible?

Mathematics Integration Level: Misc Level

A window is in the form of a rectangle above which there is a semicircle. If the perimeter of the window is p cm, show that the window will admit maximum possible light only when the radius of semicircle is cm

Mathematics Integration Level: Misc Level

Find the point on the parabola y² = 4x which is nearest to the point (2,-8)