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13 problems match “piecewise functions”. Clear

Piecewise function with equal values ●●●○○ 50 ptsLet $a \neq 0$ be a real number, and define the function $f(x)$ by $f(x)=\begin{cases}2x+a, & x<1,\\ -x-2a, & x\geq 1.\end{cases}$ If $f(1-a)=f(1+a)$ , find the value of $a$ . Give your answer as a decimal.
Alternating sum of minimum values ●●●○○ 50 ptsFor a positive integer $n$ , define $f_n(x)=\left(\sum_{i=1}^{n}|x-i|\right)^2-\sum_{i=1}^{n}(x-i)^2$ on the interval $[0,n+1]$ . Let $a_n$ be the minimum value of $f_n(x)$ . Find $\sum_{n=1}^{11}(-1)^{n+1}a_n$ .
Smallest n with many solutions ●●●●○ 100 ptsLet $f(x)=|2\{x\}-1|$ , where $\{x\}$ denotes the fractional part of $x$ . Find the smallest positive integer $n$ such that the equation $n f(x f(x))=x$ has at least $2012$ real solutions.