Step 1: Read the structure
The skeleton has an OH on one carbon, and a C=C bond next to it. On the left of the double bond is a CH carrying a methyl group and an ethyl group. On the right of the OH carbon is a CH carrying a methyl group and an ethyl group.
Step 2: Find the parent chain
The longest chain that contains both the OH carbon and the double bond runs through both ethyl groups: \(\text{CH}_3\text{CH}_2-\text{CH}(\text{CH}_3)-\text{CH=CH}-\text{CH(OH)}-\text{CH}(\text{CH}_3)-\text{CH}_2\text{CH}_3\). It has 9 carbons, so the parent is nonene.
Step 3: Number the chain
The -OH group has the first priority for numbering. Numbering from the right end gives C1 = \(\text{CH}_3\), C2 = \(\text{CH}_2\), C3 = \(\text{CH}(\text{CH}_3)\), C4 = \(\text{CH(OH)}\), C5 = \(\text{CH=}\), C6 = \(=\text{CH}\), C7 = \(\text{CH}(\text{CH}_3)\), C8, C9. Numbering from the left would put OH on C6, which is higher.
Step 4: Assemble the name
OH at C4, double bond at C5, methyl groups at C3 and C7. The name is 3,7-dimethylnon-5-en-4-ol. The other names choose a shorter chain or give wrong locants.
Final Answer:
The correct name is 3,7-dimethylnon-5-en-4-ol. This is option (C).
\[ \boxed{\text{(C) }\text{3,7-Dimethylnon-5-en-4-ol}} \]
