Question

Why is f(u)=u^2 monotonic for positive u, but not for negative u? When you square the negative u the output keeps increasing, so why wouldn't it also be monotonic?

Answer 1

The square is a monotonic function on the positive number line interval. On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function.

in the positive number line the absolute value increases number by number i.e. 1<2<3<4<5<.... so their squares also follow the monotonic pattern i.e. 1<4<9<16<25...

but in case on negative number line the value decreases number by number i.e. -1>-2>-3>-4>-5... as the are in decreasing order but their squares will be in ascending order because the square of the negative number will not stay negative.