Logic Tip: Don't fall for the trap of treating the methyl group as a substituent (Options A and D). Always find the absolutely longest continuous chain first. $CH_3-CH(.......)- .......$ is not a substituted butane; it is the start of a pentane chain.
Concept:
When assigning an IUPAC name to an organic compound with multiple functional groups, we must follow the priority rules. The principal functional group dictates the suffix of the name and gets the lowest possible locant (number) when numbering the carbon chain. The amine group ($-NH_2$) has a higher priority than the alkene double bond ($-C=C-$).
Step 1: Identify the longest continuous carbon chain and the principal functional group.
The longest carbon chain containing both the amine group and the double bond has 5 carbon atoms. Therefore, the parent alkane is "pentane".
Because of the double bond, the primary chain becomes "pentene".
Because of the amine group, the principal functional group suffix is "-amine".
Combining these, the base name format will be pent-x-en-y-amine.
Step 2: Number the carbon chain to give the lowest locant to the principal group.
We can number the chain from left-to-right or right-to-left. Since the amine group has higher priority, it must get the lower number.
Numbering from left to right:
$^1\text{CH}_3-^2\text{CH}(\text{NH}_2)-^3\text{CH}=^4\text{CH}-^5\text{CH}_3$
In this scheme, the amine is at position 2, and the double bond starts at position 3.
Numbering from right to left:
$^5\text{CH}_3-^4\text{CH}(\text{NH}_2)-^3\text{CH}=^2\text{CH}-^1\text{CH}_3$
In this scheme, the amine is at position 4, and the double bond starts at position 2.
Comparing the two, the left-to-right numbering gives the principal functional group (amine) a lower number (2 vs 4). Therefore, the correct numbering is from left to right.
Step 3: Assemble the final IUPAC name.
Using the chosen locants:
Parent chain: pent
Double bond position: 3-en
Amine position: 2-amine
Putting it all together, dropping the 'e' from 'ene' before the vowel 'a':
Pent-3-en-2-amine