(3) A line of length ${\displaystyle p}$ is to be cut up into ${\displaystyle 4}$ parts and put together as a rectangle. Show that the area of the rectangle will be a maximum if each of its sides is equal to ${\displaystyle {\tfrac {1}{4}}p}$.

(4) A piece of string ${\displaystyle 30}$ inches long has its two ends joined together and is stretched by ${\displaystyle 3}$ pegs so as to form a triangle. What is the largest triangular area that can be enclosed by the string?

(5) Plot the curve corresponding to the equation

also find ${\displaystyle {\dfrac {dy}{dx}}}$, and deduce the value of ${\displaystyle x}$ that will make ${\displaystyle y}$ a minimum; and find that minimum value of ${\displaystyle y}$.

(6) If ${\displaystyle y=x^{5}-5x}$, find what values of ${\displaystyle x}$ will make ${\displaystyle y}$ a maximum or a minimum.

(7) What is the smallest square that can be inscribed in a given square?

(8) Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.

(9) Inscribe in a sphere, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.