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https://colab.research.google.com/drive/1Pdg_FyY5qGkP-_Uy0pIO8HUD9eyEKJp8
https://colab.research.google.com/drive/13RbnMbhb99tzA52rIClSTZ1V8sPB0Z7b

Yeah – you can do it.

A question about convex function

I have the following question,
Given a real number $x$ and a positive integer $m$.
For $k\in \{1,2,\cdots,m\}$. Let $\phi_k$ be a real-valued function defined on $k$-th segment of interval $[0,x]$. Let $$\phi(x)=\frac{1}{m}\sum_{i=1}^m\phi_i(x).$$
I have to show $\phi(x)$ is convex on $[0,x]$ in term of $\phi_k(x)$.
First I tried to divide $\phi(x)$ into $m$ parts, that is,
$$\phi(x)=\sum_{i=1}^m\phi_i(x)=\sum_{k=1}^m\phi_k(x) + \sum_{k=1}^m(\phi_k(x)-\phi_k(0)).$$
Next, I tried to use the inequality of arithmetic and geometric means to show $$\sum_{k=1}^m(\phi_k(x)-\phi_k(0))\le 0.$$
Any help would be appreciated.

A:

Hint:
$f(x)=\sum\limits_{k=1}^{m}\phi_{k}(x)$ is increasing
$g(x)=x^{2}$ is convex on $\mathbb{R}$ and all its derivative is non-negative, and $g(x)-g(0)=x^{2}-0=(x+0)^{2}-0=(x+0)^{2}\geq0$.
$f(x)-f(0)=\sum\limits_{k=1}^{m}(\phi_{k}(x)-\phi_{k}(0))\geq0$.
so $f(x)$ is convex.

Q:

Is it possible to use a function as a prototype in TypeScript?

I’ve been looking at the TypeScript documentation and there’s a
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Date: August 1, 2022

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