If the solve the problem

Given: $f(x)=\sin 2 x+5$

We know that $-1 \leq \sin 2 x \leq 1$

$\Rightarrow-1+5 \leq \sin 2 x+5 \leq 1+5$

$\Rightarrow 4 \leq \sin 2 x+5 \leq 6$

$\Rightarrow 4 \leq f(x) \leq 6$

Hence, the maximum and minimum values of $f$ are 6 and 4 , respectively.