Product of increasing functions need not be increasing
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 .
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.