The so-called law of diminishing returns in economics
expresses the idea that function values that are
increasing can sometimes go up fast, and sometimes
go up slow. If we have a profit function and know
that increasing production will increase profit, that
investment may be deemed \"worth it\" if the increase
in profit is large. But, if the increased profit is small, it
may be better to invest elsewhere.
The Point of diminishing returns is the point where the increasing function values are starting to slow. In other words, it is
when the function is increasing ($f'(x) > 0$) and when concavity switches from positive to negative, i.e. an inflection point.
8. Let $x > 0$. Find the point of diminishing returns for $P(x) = 56x^{4/3} - x^{7/3}$
9. Does $P(x) = x^2 \ln x$ have a point of diminishing returns? Why or why not?
