We know, DCFL is a subset of CFL.
For a language to be DCFL, it has to be accepted by a DPDA (Deterministic Pushdown Automata) i.e. for every input symbol and stack symbol, atmost one possible move.
For DCFL: 1) Machine can predict what to do next without guessing.
2) Strings can be parse left to right unambigiously.
For non-DCFL: 1) Machine need to guess a point or symbol.
2) The grammar or parse tree is ambigious or requires epsillion-transitions that lead to non determinism.
Analysing each language, L1 = {wcw | w belongs to {a,b}*}, in the given language -----------c----------------- no of characters before c = no of characters after c. Since machine has to remember th previous one therefore it cannot be context free language. If its not CFL, there's no question of DCFL and NON DCFL.
Analysing other langiage, L2 = { wcw^R|w belongs to as pervious one}, in the given language R claraly states that after c there has to be the reverse order of the w. The machine remembers w on the stack; it only needs guessing if the separator is not unambiguously marked., and, as it has to remember the point about which the reverse must happen, it is deterministic CFL.
Analysing another option, L3 = {ww^R| w belongs to as previous one} in the given language there's no point about which machine knows wether to stop popping or start therefore it is not a DCFL. But it is CFL.
