1. Given function

We need the domain of $f(x)$.

For the domain:

• the logarithm argument must be positive,
• the argument of $\cos^{-1}$ must lie in $[-1,1]$,
• and denominators must be nonzero.

2. Condition from the logarithm

We need

Factor the denominator:

So the inequality is

Critical points are:

Now check signs on intervals:

• For $x< -\frac32$, expression is negative.
• For $-\frac32 < x < -1$, expression is positive.
• For $-1 < x < \frac34$, expression is negative.
• For $x>\frac34$, expression is positive.

Hence logarithm condition gives

3. Condition from the inverse cosine

We need

We solve both inequalities.

(i) Solve

Bring to one side:

Critical points: $x=-2,-\frac13$.

Sign analysis gives

(ii) Solve

Critical points: $x=-2,3$.

Sign analysis gives

Now intersect the two results:

So the $\cos^{-1}$ condition gives

4. Intersect both domain conditions

From logarithm:

From inverse cosine:

Their intersection is

Thus,

So,

5. Compute $5\beta-4\alpha$

6. Final answer

So the correct option is B.