A Monotone Piecewise Split

★ 7/10functions Monotonicity Recursion sequences Induction

Problem

A quadratic function $f(x)=ax^2+bx+c$ satisfies the following three conditions:

1. $2$ is a zero of $f(x)$ ; 2. the maximum value of $f(x)$ is $1$ ; 3. $f(x+1)=f(1-x)$ for every real number $x$ .

Part 1. Find $f(x)$ .

Part 2. Let $A$ and $B$ be disjoint sets with $A\cup B=(0,1)$ , and suppose

g(x)=\begin{cases}x,&x\in A,\\ f(x),&x\in B\end{cases}

is an increasing function on its domain $(0,1)$ . Let $0<x_0<x'<1$ . Prove that if $x_0\in B$ , then $x'\in B$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Monotonicity, Recursion, sequences, Induction

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