Interpret the question

We need the maximum possible denticity of the given ligand toward:

• a common transition metal ion, and
• an inner-transition metal ion.

The ligand referred to here is the well-known aminopolycarboxylate ligand that can potentially bind through:

• 2 nitrogen donor atoms, and
• 6 oxygen donor atoms from carboxylate groups.

So, the ligand has a total of $2 + 6 = 8$ possible donor atoms.

For a common transition metal ion

Common transition metal ions usually prefer coordination number up to 6 in stable complexes.

Therefore, even if the ligand has 8 donor atoms available, toward a common transition metal ion it can at most behave as a: $\boxed{6\text{-dentate ligand}}$

For an inner-transition metal ion

Inner-transition metal ions (especially lanthanoids/actinoids) are larger in size and can accommodate higher coordination numbers, often 8 or more.

Hence, this ligand can use all its 8 donor atoms and act as: $\boxed{8\text{-dentate ligand}}$

Match with options

Thus, the maximum possible denticities are:

• toward a common transition metal ion: 6
• toward an inner-transition metal ion: 8

So the correct option is: $\boxed{\text{C: }6\text{ and }8}$

Comparison with stored correct answer

Stored correct answer = C.

Our derived answer = C.

Hence, they agree.