While familiarizing myself with polynomial hierarchy, I used this book which is written by Ingo Wegener. Now I'm practicing, and on page 132 I met this exercise:
Let us consider a Boolean formula $\phi$ on the variables $x_1, ...,x_n$. Each $n$ bit vector $\overline x \in \{0,1\}^n$ is a possible assignment to variables and these vectors can be naturally classified in alphabetical order. The ODD-SMALLER-SAT-DECISION problem is to determine, being given $\phi$, if the smallest assignment $x$ which is satisfactory is such that $x_n = 1$.
How to prove that this problem is part of the complexity class $\Delta_2$?