Solve any maths problem

Step by step

1. f(x) = - \frac{2 x}{\left(x^{2} + 1\right)^{2}}

   Critical points are where f′(x) = 0 or is undefined.

2. f'(x) = \frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} - \frac{2}{\left(x^{2} + 1\right)^{2}}

   Differentiate.

3. x = - \frac{\sqrt{3}}{3}, x = \frac{\sqrt{3}}{3}

   Solve f′(x) = 0.

4. f''(x) = - \frac{48 x^{3}}{\left(x^{2} + 1\right)^{4}} + \frac{24 x}{\left(x^{2} + 1\right)^{3}}

   Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

5. f''(- \frac{\sqrt{3}}{3}) = - \frac{27 \sqrt{3}}{16} \Rightarrow (- \frac{\sqrt{3}}{3}, \frac{3 \sqrt{3}}{8}) \text{ is a local maximum}

6. f''(\frac{\sqrt{3}}{3}) = \frac{27 \sqrt{3}}{16} \Rightarrow (\frac{\sqrt{3}}{3}, - \frac{3 \sqrt{3}}{8}) \text{ is a local minimum}