1. **Problem statement:** Given the derivative function $$f'(x) = \frac{-5x}{(x^2+1)^2}$$, analyze its behavior and find critical points. 2. **Recall the formula:** Critical points occur where $$f'(x) = 0$$ or where $$f'(x)$$ is undefined. 3. **Set the numerator equal to zero:** Since the denominator $$(x^2+1)^2$$ is always positive and never zero, the zeros of $$f'(x)$$ come from the numerator: $$-5x = 0$$ 4. **Solve for $$x$$:** $$x = 0$$ 5. **Check the domain:** The function $$f'(x)$$ is defined for all real $$x$$ because the denominator is never zero. 6. **Determine intervals of increase/decrease:** - For $$x < 0$$, numerator $$-5x > 0$$ (since $$x$$ is negative, $$-5x$$ is positive), so $$f'(x) > 0$$, meaning $$f(x)$$ is increasing. - For $$x > 0$$, numerator $$-5x < 0$$, so $$f'(x) < 0$$, meaning $$f(x)$$ is decreasing. 7. **Conclusion:** There is a critical point at $$x=0$$ where $$f(x)$$ changes from increasing to decreasing, indicating a local maximum at $$x=0$$. **Final answer:** The function $$f(x)$$ has a local maximum at $$x=0$$.