Don't to do this problem. Very boring and you will learn nothing from it.
If the first advise does not work:
- First guess what is the answer,
- Second prove that it is a strict local maximum in the space of relative orientations of these two bodies; it is essentially SO(3). (This part should be nearly a calculus problem.)
- Finally, show that it is global maximum by estimating the value at a $\varepsilon$-net of SO(3) for small enough $\varepsilon$. (While doing this you may also learn that your guess was wrong and it will suggest a better guess.)
Minkowski subtraction might help to do the last two steps.
The computers now are better than in 80's so all this might work.