Product of increasing functions need not be increasing
Contents
Statement
It is possible to have two real-valued functions , both defined on all real numbers, such that both
and
are increasing functions but the product
(defined as
) is not an increasing function on all of
.
We can also adapt this to functions defined on any open interval or closed interval instead of all real numbers.
Proof
Counterexample
Since we need to disprove a general statement, it is enough to exhibit one counterexample.
The example is with both the identity function:
These are both increasing on all real numbers.
The product is:
This is the square function which is decreasing on and increasing on
.
Generic idea behind counterexample
A product of negative increasing functions is decreasing.
Compatibility with product rule for differentiation
The product rule for differentiation states that:
In particular, the sign of depends not merely on the signs of
and
but also on the signs of
and
. In particular, if
is negative, then multiplying by a positive
gives a negative product. This is why
can be negative even though
are both positive.