To simplify the given Boolean function f = A.B.C.D + A.C.D + A.D, let's apply Boolean algebra and simplify each term:

f = A.B.C.D + A.C.D + A.D

First, let's factor out the common term 'A.D' from each term:

f = A.D(B.C + C + 1)

Now, let's simplify the expression (B.C + C + 1):

B.C + C + 1 = C(B + 1) + 1 = C + 1

Substituting this back into the original expression:

f = A.D(C + 1)

Now, we can see that the simplified form of the Boolean function f is A.D.

Therefore, the correct answer is A) A.D.

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