Function Fundamentals

Algebra Level 2

$f(x), g(x), h(x)$ are three functions defined at $\mathbb R$ .

If $f(x)+g(x), f(x)+h(x), g(x)+h(x)$ are all strictly increasing, is it always true that at least one of $f(x), g(x), h(x)$ is strictly increasing？

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1 solution

Patrick Corn
Mar 3, 2020

We can even make the functions continuous. For any real number $a,$ let $m_a(x) = \begin{cases} x & \text{if } x \le a \\ \frac{3a-x}2 & \text{if } a \le x \le a+2 \\ x-3 & \text{if } a+2 \le x \end{cases}$

Let $f(x) = m_0(x), g(x) = m_2(x), h(x) = m_4(x).$ Then $f+g, f+h,$ and $g+h$ are all strictly increasing; they're piecewise linear functions, and each piece has slope either $2$ or $1/2.$ But none of $f,g,h$ are increasing.

I could not post a solution because I can't write latex code for the definition of the function $m_a(x)$ in the style you used. What is the code please.

A Former Brilliant Member - 1 year, 3 months ago

Function Fundamentals

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