Calculus: Problems of Maxima and Minima

A number of non-sketched problems can be solved using calculus. This is usually used in optimisation problems. For example, maximising profit, storage, consumption or size of an area.


For example:

The equations for the expense per year (in units of ten thousand dollars) of running a certain business is given by E={x}^{2}-6x+12, where x is the number (in hundreds) of items manufactured.

a) Find the expense of running the business if no items are manufactured.

b) Find the number of items needed to minimise the expense of the business.

c) Find the minimum expense of the business.


Treat the following problem with your knowledge about derivatives.

a) When x=0,


So the expense of running the business when no items are manufactured is 12 x $10 000, or $120 000 per year.


b) \cfrac{dE}{dx}=2x-6

For stationary points,

\cfrac{dE}{dx}=0 \\ \\ 2x-6=0 \\ 2x=6 \\ x=3 \\ \\ \cfrac{{d}^{2}E}{d{x}^{2}}=2>0

Since the second derivative is concave upwards, then x=3 gives a minimum value.

So 300 items manufactured each year will give the minimum expenses.


c) When x=3,


So the minimum expense per year is $30 000.


Other problems require you to derive the function of the problems on your own.

For example:

ABCD is a rectangle with AB = 10cm and BC = 8cm. Length AEx cm and CFy cm.


a) Show that triangles AEB and CBF are similar.

b) Show that xy = 80.

c) Show that triangle EDF has area given by A=80+5x+\cfrac{320}{x}.


Other examples: