Show that the reactangle of maximum perimeter which can be inscribed in a circle of radius a is a square of side `sqrt2 a.`

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

Shows that the triangle of greatest area that can be inscribed in a circle is equilateral.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is (2R)/(sqrt(3)) . Find the volume of the largest cylinder inscribed in a sphere of radius R.

An isosceles triangle of vertical angle 2 theta is inscribed in a circle of radius a. Show that the area of triangle is maximum when theta = (pi)/(6) .

Show that all the rectangles with a given perimeter, the square has the largest area.