Step 1: Understanding the Question:
The question tests the concept of synergic bonding in organometallic compounds, specifically metal carbonyls.
Step 2: Detailed Explanation:
Statement A: Bonding in metal carbonyls involves \(\sigma\)-donation from CO to the metal and \(\pi\)-back-donation from the metal to CO. Thus, the bond has both \(\sigma\) and \(\pi\) character. Correct.
Statement B: Synergic bonding strengthens the bond between the metal and the carbon (M-C bond) while weakening the C-O bond. The statement says the M-C bond becomes weak, which is Incorrect.
Statement C: The \(\sigma\)-bond is indeed formed by the lone pair of Carbon (from its HOMO) being donated to an empty orbital of the metal. Correct.
Statement D: The \(\pi\)-bond is formed by the back-donation of electrons from a filled metal d-orbital into the vacant anti-bonding \(\pi^*\) orbital (LUMO) of the CO ligand. Correct.
Step 3: Final Answer:
Statements A, C, and D are correct.