**Problem : **
Find the critical points and inflection points of the function *f* (*x*) = *x*^{4} -2*x*^{2} (with domain
the set of all real numbers). Which of the critical points are local minima? local
maxima? Is there a global minimum or maximum?

f'(x) | = | 4x^{3} - 4x | |

= | 4(x + 1)x(x - 1) | ||

f''(x) | = | 12x^{2} - 4 | |

= | 4(3x^{2} - 1) |

We see that

f''(- 1) | = | 8 | |

f''(0) | = | -4 | |

f''(1) | = | 8 |

so by the second derivative test,

f (- 1) | = | -1 | |

f (0) | = | 0 | |

f (1) | = | -1 |

so

Figure %: Graph of *f* (*x*) = *x*^{4} -2*x*^{2}

**Problem : **
Find the inflection points of *f* (*x*) = *e*^{-x2}. (This famous function
is called the *gaussian.*)

f'(x) | = | -2xe^{-x2}, | |

f''(x) | = | (- 2x)(- 2xe^{-x2}) + (- 2)(e^{-x2}) | |

= | (4x^{2} -2)e^{-x2} |

Solving

Figure %: Graph of *f* (*x*) = e^{-x2}

**Problem : **
Find a function *f* (*x*) with inflection points at *x* = 1 and *x* = 2.

f''(x) = (x - 1)(x - 2) = x^{2} - 3x + 2 |

We search for a function with this derivative, ultimately obtaining

f'(x) = - + 2x |

Similarly, a function with this derivative is given by

f (x) = - + x^{2} |

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