There are three different hypotheses proposed by scientists for explaining species richness in the tropics.
(1) Tropical latitudes receive more solar energy than temperate regions which leads to high productivity and high species diversity.
(2) Tropical regions have less seasonal variations and have a more or less constant environment. This promotes the niche specialization and thus high species richness.
(3) Temperate regions were subjected to glaciations during the ice age while tropical regions remained undisturbed which led to an increase in the species diversity in this region.
Answer the following questions:
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
On the basis of the above information, answer the following questions :
Find \( \frac{dS}{dx} \).

Ecologists discovered two broad kinds of diversity patterns, namely: