Mammalian $\beta$-oxidation occurs in the mitochondrion, whereas fatty acid biosynthesis is cytoplasmic.
- The rate-limiting in mammalian fatty acid oxidation is considered to be carnitine palmitoyltransferase (CAT), a key enzyme in the transport of acetyl-CoA across the inner mitochondrial membrane. This enzyme is inhibited by low micromolar concentrations of malonyl-CoA (see Beta oxidation of fatty acids by H. Schulz).
Malonyl-CoA, of course, is a key intermediate in fatty acid biosynthesis. Salih Wakil showed that CO2 is required for fatty acid biosynthesis, but that carbon atoms from CO2 do not appear in fatty acids, and it is now known that this is due to the formation of .malonyl-CoA by acetyI-CoA carboxylase, the rate-limiting reaction of fatty acid synthesis.
- Thus when fatty acid biosynthesis is taking place, the concentration of malonyl-CoA is high and CAT (and therefore $\beta$-oxidation) is inhibited (Schulz, 1991).
However, as the OP points out, micromolar concentrations of acetyl-CoA inhibit 3-ketoacyI-CoA thiolase when the concentration of CoASH is low, and the Acetyl-CoA/CoASH ratio within the mitochondrion may also play a role in the regulation of beta-oxidation (Schulz, 1991).
As the OP says, it is a debatable point, and it seems to me, an example of a badly worded exam question. I think the 'expected' answer is probably malonyl-CoA, and there is no doubt that this molecule plays a key role in the regulation of both fatty acid oxidation and fatty acid biosynthesis. Nevertheless, the inhibition of thiolase by Acetyl-CoA may also play a role in the regulation of beta-oxidation.
Perhaps your Professor is an example of the type that Warburg told Hans Krebs to attach himself to? :-)