(6) Verify that the sum of three quantities $x$,~$y$,~$z$, whose product is a constant~$k$, is maximum when these three quantities are equal.

(7) Find the maximum or minimum of the function \[ u = x + 2xy + y. \] (7)~ Minimum for $x = y = -\frac{1}{2}$.

(8) The post-office regulations state that no parcel is to be of such a size that its length plus its girth exceeds $6$~feet. What is the greatest volume that can be sent by post ({a})~in the case of a package of rectangular cross section; ({b})~in the case of a package of circular cross section. (8)~ ({a}) Length $2$~feet, width = depth = $1$~foot, vol.\ = $2$~cubic feet. ({b}) Radius = $\frac{2}{\pi}$ feet = $7.46$~in., length = $2$~feet, vol.\ = $2.54$.

(9) Divide $\pi$ into $3$~parts such that the continued product of their sines may be a maximum or minimum. (9)~ All three parts equal; the product is maximum.

(10) Find the maximum or minimum of $u = \frac{e^{x+y}}{xy}$. (10)~ Minimum for $x = y = 1$.

(11) Find maximum and minimum of \[ u = y + 2x - 2 \ln y - \ln x. \] (11)~ Min.: $x = \frac{1}{2}$ and $y = 2$.

(12) A telpherage bucket of given capacity has the shape of a horizontal isosceles triangular prism with the apex underneath, and the opposite face open. Find its dimensions in order that the least amount of iron sheet may be used in its construction. (12)~ Angle at apex $= 90^\circ $; equal sides = length = $\sqrt[3]{2V}$.