If A and B are candidate keys, then AB is candidate key or not

If you're talking about identifying candidate keys for relational synthesis you need to know the dependencies. If you have the functional dependencies you can use the relational synthesis algorithm; a synopsis of which can be found here.

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A superkey (and therefore a candidate key) that does not contain all attributes must contain at least an attribute set that is the left side of a functional dependency. Otherwise it would not be possible to generate an additional attribute from the attribute set of this superkey and this contradicts the definition of a superkey.

But a subset of ABCDE containing at least one of C,AB,AE or DE is already a superkey and to be minimal it is necessary that it contains only one of these subsets.

So the only subsets that must me investigated are that that are supersets of AC. But a superset of AC cannot be a minimal superkey because C already is a key.

Therefore the only candidate keys are C,AB,AE and DE