$T=\frac{2\pi }{\left| b \right|}\xrightarrow{T=\frac{1}{2}}\frac{2\pi }{\left| b\pi  \right|}=\frac{1}{2}\Rightarrow \frac{2}{\left| b \right|}=\frac{1}{2}\Rightarrow \left| b \right|=4\Rightarrow b=\pm 4$  $f(x)=0\Rightarrow a\operatorname{Sin}b\pi x=0\Rightarrow a\operatorname{Sin}(\pm 4\pi x)=0\Rightarrow \pm a\operatorname{Sin}4\pi x=0\Rightarrow 4\pi x=k\pi \Rightarrow 4x=k\Rightarrow x=\frac{k}{4}$  $x\in \left[ 0,1 \right]\Rightarrow 0\le x\le 1\Rightarrow 0\le \frac{k}{4}\le 1\Rightarrow 0\le x\le 4\xrightarrow{k\in Z}k\in \left\{ 0,1,2,3,4 \right\}$