find maximum value and minimum value

Question

$y=x^2$ and $D_f=(0,2]$.

So, my interval is $0<x^2<=2$

But, according to my book it has no minimum values. But, it has maximum value which is $2^2=4$

According to me, it should have minimum value either. Cause, $1$ exists in that interval also.So, $1$ is minimum value. But, why they said that there's no minimum value?

Answer 1

This function is increasing on this interval (prove it).

Suppose the minimum occured at $x > 0$. Consider $x/2$ - uh-oh.

See the problem? No minimum exists on $(0, 2].$