1. **State the problem:** We have a function $$f(x) = 3ax^3 - bx - 5$$ and it is given that there is a local maximum at $$x=1$$. 2. **Recall the condition for local maxima:** At a local maximum, the first derivative $$f'(x)$$ must be zero. 3. **Find the first derivative:** $$f'(x) = \frac{d}{dx}(3ax^3 - bx - 5) = 9ax^2 - b$$ 4. **Apply the condition at $$x=1$$:** $$f'(1) = 9a(1)^2 - b = 9a - b = 0$$ 5. **Solve for $$b$$ in terms of $$a$$:** $$9a - b = 0 \implies b = 9a$$ 6. **Find the ratio $$\frac{b}{a}$$:** $$\frac{b}{a} = \frac{9a}{a} = 9$$ 7. **Answer:** The ratio $$\frac{b}{a} = 9$$, which corresponds to option (b).