The correct IUPAC name of the following compound is:

Based on IUPAC nomenclature rules, the principal functional group must be given the lowest possible number. Here, the ketone ($>C=O$) group has higher priority than the carbon-carbon double bond ($-C=C-$) [Source 153]. The longest continuous carbon chain containing both the ketone group and the double bond is selected. If the structure is $CH_3-CO-CH=C(CH_3)-CH_2-CH_3$ (often drawn misleadingly as a 5-carbon chain with an ethyl substituent), the longest chain actually contains 6 carbons (hex). Numbering from the end closer to the carbonyl group gives the ketone at position 2, the alkene at position 3, and a methyl group substituent at position 4. Hence, the correct IUPAC name is 4-Methylhex-3-en-2-one.