# Properties of Convex Functions

Here we will talk about properties of convex (or concave upward) function.

We already noted that if function `f(x)` is concave upward then `-f(x)` is concave downward. So, these properties also hold for concave downward functions.

**Property 1**. Suppose `f(x)` is concave upward and `c>0` is arbitrary constant then `cf(x)` is concave upward.

**Property 2**. Sum of any number of concave upward functions is concave upward.

Note, that this doesn't hold for product, i.e. product of two concave upward functions can be not concave upward.

**Property 3**. Suppose that we are given two functions `y=f(u)` and `u=g(x)`. Following is true for composite function `f(g(x))`:

`f(u)` | `g(x)` | `f(g(x))` |

concave upward, increasing | concave upward | concave upward |

concave upward, decreasing | concave downward | concave upward |

concave downward, increasing | concave downward | concave downward |

concave downward, decreasing | concave upward | concave downward |

**Property 4**. If for the function `y=f(x)` there exists unique inverse `f^(-1)(x)` then the following is true

`f(x)` | `f^(-1)(x)` |

concave upward, increasing | concave downward, increasing |

concave upward, decreasing | concave upward, decreasing |

concave downward, increasing | concave upward, increasing |

concave downward, decreasing | concave downward, decreasing |

**Property 5**. Non-constant concave upward on interval `X` function `f(x)` can't attain global maximum inside this interval. In other words global maximum can be only at one of the endpoints.