1. **State the problem:** We need to find the intervals where the function $$f(x) = -x^3 + 12x^2 - 36x$$ is increasing. 2. **Recall the rule:** A function is increasing where its derivative $$f'(x)$$ is positive. 3. **Find the derivative:** $$f'(x) = \frac{d}{dx}(-x^3 + 12x^2 - 36x) = -3x^2 + 24x - 36$$ 4. **Set the derivative greater than zero to find increasing intervals:** $$-3x^2 + 24x - 36 > 0$$ 5. **Divide the inequality by -3 (note this reverses the inequality sign):** $$x^2 - 8x + 12 < 0$$ 6. **Factor the quadratic:** $$x^2 - 8x + 12 = (x - 6)(x - 2)$$ 7. **Solve the inequality:** $$(x - 6)(x - 2) < 0$$ This product is less than zero when $$x$$ is between the roots: $$2 < x < 6$$ 8. **Conclusion:** The function $$f(x)$$ is increasing on the interval $$\boxed{(2, 6)}$$.