1. **State the problem:** We need to find the intervals where the function $f(x) = x^3 - 21x$ is increasing. 2. **Recall the rule:** A function is increasing where its derivative $f'(x)$ is positive. 3. **Find the derivative:** $$f'(x) = \frac{d}{dx}(x^3 - 21x) = 3x^2 - 21$$ 4. **Set the derivative greater than zero to find increasing intervals:** $$3x^2 - 21 > 0$$ 5. **Solve the inequality:** $$3x^2 > 21$$ $$x^2 > 7$$ $$x > \sqrt{7} \quad \text{or} \quad x < -\sqrt{7}$$ 6. **Conclusion:** The function $f(x)$ is increasing on the intervals: $$(-\infty, -\sqrt{7}) \cup (\sqrt{7}, \infty)$$