What is the correct name of 3-isopropyl-4-methylhexane?
Both the names you have given are not the preferred name for the molecule above. You are correct that the preferred name maximises the number of substituents when there is a choice: see Alkane nomenclature - choosing the parent chain - maximize or minimize substituents? So, a name of the form a-ethyl-(b,c)-dimethylhexane (where a, b, and c are appropriately chosen locant numbers) is preferred over a name of the form a-isopropyl-b-methylhexane, as it has three substituents as opposed to two.

However, the numbering on the right is not correct either. If you number the carbon chain the other way around instead, you would obtain the name 3-ethyl-2,4-dimethylhexane:



This is preferred over your proposed name 4-ethyl-3,5-dimethylhexane, because the locant set (2,3,4) in 3-ethyl-2,4-dimethylhexane is lower than the locant set (3,4,5) in 4-ethyl-3,5-dimethylhexane.

Note also that according to current IUPAC nomenclature recommendations (P-57.1.1.1 of the 2013 Blue Book), the preferred prefix for $\ce{-CH(CH3)2}$ is "propan-2-yl", not "isopropyl".