Can arguments of max() function be factored out when multiplied?

Question

$$\max(x_1-k_1,0)^2=\max(x_1-k_1,0)\max(x_1-k_1,0)$$

Is the following true?

$$\max(x_1-k_1,0)\max(x_1-k_1,0)=(x_1-k_1)\max(x_1-k_1,0)$$

And more generally...

$$\max(x_1-k_1,0)\max(x_2-k_2,0)=(x_2-k_2)\max(x_1-k_1,0)$$

?

Answer 1

What you have outlined is incorrect - Let us simplify the problem $$ \max(k, 0)^2 $$ is what you want to find. If we assume what you say $$ \max(k, 0)\max(k, 0) = k\max(k, 0) $$ then it is clear you have already assumed that $\max(k, 0) = k$ so why not the other? since they are the same. Lets look at the cases $$ \max(k, 0)\max(k, 0) = \cases{k^2 \;\; k >0 \\ 0 \;\; k \leq 0} $$ this suggests that we have $$ \max(k, 0)\max(k, 0) = \max(k^2, 0) $$ now the general case $$ \max(k, 0)\max(m, 0) = \cases{km \;\; k > 0, m > 0\\ k\cdot 0 \;\;k > 0, m \leq 0\\ m \cdot 0 \;\;k \leq 0, m > 0\\ 0 \;\;k \leq 0, m \leq 0} $$ can you work out a max of the above.