1. **Problem statement:** Find the monotony intervals of the function $$f(x) = (x + 13) |x + 13|$$. 2. **Rewrite the function:** The absolute value function splits into cases: - For $$x + 13 \geq 0$$ (i.e., $$x \geq -13$$), $$|x + 13| = x + 13$$, so $$f(x) = (x + 13)(x + 13) = (x + 13)^2$$. - For $$x + 13 < 0$$ (i.e., $$x < -13$$), $$|x + 13| = -(x + 13)$$, so $$f(x) = (x + 13)(-(x + 13)) = -(x + 13)^2$$. 3. **Find the derivative in each interval:** - For $$x > -13$$: $$f(x) = (x + 13)^2$$ $$f'(x) = 2(x + 13)$$ - For $$x < -13$$: $$f(x) = -(x + 13)^2$$ $$f'(x) = -2(x + 13)$$ 4. **Analyze the sign of the derivative:** - For $$x > -13$$, $$f'(x) = 2(x + 13) > 0$$, so $$f$$ is increasing on $$(-13, \infty)$$. - For $$x < -13$$, $$f'(x) = -2(x + 13)$$. Since $$x + 13 < 0$$, $$-2(x + 13) > 0$$, so $$f$$ is also increasing on $$(-\infty, -13)$$. 5. **At $$x = -13$$, the function is continuous but the derivative changes form.** 6. **Conclusion:** The function is increasing on $$\mathbb{R}$$ except possibly at $$x = -13$$ where the derivative is not defined (due to the absolute value). **Answer:** The function is increasing over $$\mathbb{R}$$.