The minimum value that $f\left(x\right)=4x^2-4x+11+\sin 3\pi x$  is 

We have,

$\begin{array}{l}f\left(x\right)=4x^2-4x+11+\sin 3\pi x\\ \Rightarrow f\left(x\right)=4{\left(x-\frac{1}{2}\right)}^2+10+\sin 3\pi x\end{array}$

We observe that $4{\left(x-\frac{1}{2}\right)}^2$ takes its least value $0$ at $x=\frac{1}{2}$ and

at this point $\sin 3\pi x$ also takes its least value -1.

Hence, $f \left(x\right)$ attains its least value 9 at $x=\frac{1}{2}$.