Zeros of a Piecewise Cubic

Problem

Define

f(x)=\begin{cases}x,&x\geqslant a,\\ x^3-3x,&x<a.\end{cases}

1. If $f(x)$ has exactly two zeros, find the range of the real number $a$ . 2. If $a\leqslant -2$ , find all $x$ satisfying $f(x)+f(x-1)>-3$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Monotonicity, Casework

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