Solve any maths problem

Step by step

1. f(x) = - 2 x e^{- x^{2}}

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

2. f'(x) = 4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}

   Differentiate.

3. x = - \frac{\sqrt{2}}{2}, x = \frac{\sqrt{2}}{2}

   Solve f′(x) = 0.

4. f''(x) = - 8 x^{3} e^{- x^{2}} + 12 x e^{- x^{2}}

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

5. f''(- \frac{\sqrt{2}}{2}) = - \frac{4 \sqrt{2}}{e^{\frac{1}{2}}} \Rightarrow (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{e^{\frac{1}{2}}}) \text{ is a local maximum}

6. f''(\frac{\sqrt{2}}{2}) = \frac{4 \sqrt{2}}{e^{\frac{1}{2}}} \Rightarrow (\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{e^{\frac{1}{2}}}) \text{ is a local minimum}