# Properties of Convex Functions

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

We already noted that if function $$${f{{\left({x}\right)}}}$$$ is concave upward then $$$-{f{{\left({x}\right)}}}$$$ is concave downward. So, these properties also hold for concave downward functions.

**Property 1.** Suppose $$${f{{\left({x}\right)}}}$$$ is concave upward and $$${c}>{0}$$$ is arbitrary constant then $$${c}{f{{\left({x}\right)}}}$$$ 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 $$${y}={f{{\left({u}\right)}}}$$$ and $$${u}={g{{\left({x}\right)}}}$$$. Following is true for composite function $$${f{{\left({g{{\left({x}\right)}}}\right)}}}$$$:

$$${f{{\left({u}\right)}}}$$$ | $$${g{{\left({x}\right)}}}$$$ | $$${f{{\left({g{{\left({x}\right)}}}\right)}}}$$$ |

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 $$${y}={f{{\left({x}\right)}}}$$$ there exists unique inverse $$${{f}}^{{-{1}}}{\left({x}\right)}$$$ then the following is true

$$${f{{\left({x}\right)}}}$$$ | $$${{f}}^{{-{1}}}{\left({x}\right)}$$$ |

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 $$${f{{\left({x}\right)}}}$$$ can't attain global maximum inside this interval. In other words global maximum can be only at one of the endpoints.