in my textbook, while the author trying to prove that if $f$ is an integrable function on a subrectangle $S$ which is a part of a region in $\mathbb R^2$ let's call it $R$, Then we have $$\int_Rk\cdot f=k\int_Rf, \text{ where } k\in \mathbb R$$
he said that $$\min_{v\in S}(k\cdot f(v))=k\cdot \min_{v\in S}(f(v))$$ but how can I prove this, the author didn't proved it, is it obvious?
This is not true unless $k\geq 0.$
For example, if $f(x,y)=\sin^2 x\sin^2 y$ on $S=[0,2\pi]\times[0,2\pi],$ and $k=-1$ then $$-1=\min_{v\in S} kf(v)\neq k\min_{v\in S}f(v)=k\cdot 0=0.$$
When $k<0$ you have:
$$\min_{v\in S} k\cdot f(v) = k\cdot \max_{v\in S} f(v)$$
Proof for $k\geq 0:$
If $k\geq 0$ and $v_0\in S$ such that $f(v_0)=\min_{v\in S}f(v),$ which is equivalent to $$\forall v\in S, f(v_0)\leq f(v)$$ which implies that $$\forall v\in S, k\cdot f(v_0)\leq k\cdot f(v).$$