Extrema

The word “extrema” means “maxima and/or minima”. In turn, maxima is the plural of maximum, and minima is the plural of minimum. So this section is about maxima and minima.

Consider the following data about a function f(x).

– the input variable is a human being, i.e. x stands for a human being who lives on planet earth.

– the output of f is the height of the human being, measured in cm.

– the tallest thirteen year old girl in the city of Palo Alto in N. America has a height of 175 cm (about 5’9″).

– the tallest thirteen year old girl in North America has a height of 190 cm (about 6’3″).

– the tallest thirteen year old girl in the Europe/Asia combined continent has a height of 200 cm (about 6’8″).

– the tallest thirteen year old girl in South America is shorter than the tallest thirteen year old girl in Asia.

– the tallest thirteen year old girl in Africa has a height of 199 cm.

– the tallest thirteen year old girl in Australia has a height of 198 cm.

– the tallest thirteen year old girl in Antarctica has a height of 165 cm.

– the tallest thirteen year old girl in islands other than the named continents has a height of 173 cm.

– the shortest thirteen year old girl in Palo Alto has a height of 123 cm.

– the shortest thirteen year old girl in North America has a height of 123 cm.

1. What is the full domain of f? (Do you know approximately how big this domain is?)

2. Is f(x) a bounded function? If so, what are the bounds of f(x)?

3. From the above data, give an example of a local maximum of f(x). Specify what the value of the local maximum is, and where it occurs, as best you can.

4. In your specific example of a local maximum of f(x) in problem 3, what is the subdomain where the local maximum occurs?

5. In the above data, if the domain is restricted to thirteen year old girls, does f(x) have an absolute maximum?