Critical Thinking / Discussion / Writing

111. f has a relative maximum (minimum) at $x = a$ $x = a$ if there is an interval $[a, x_{1})$ $[a, x_{1})$ with $a < x_{1} < b$ $a < x_{1} < b$ such that $f (a) \geq f (x)$ $f (a) \geq f (x)$ $(f (a) \leq f (x))$ $(f (a) \leq f (x))$ for every x in the interval.
113.
1. $f (x) = x$ $f (x) = x$ on the interval $[- 1, 1]$ $[- 1, 1]$
2. $f (x) = {\begin{aligned} x & if 0 \leq x < 1 \\ 0 & if x = 1 \end{aligned}$ $f (x) = {\begin{aligned} x & if 0 \leq x < 1 \\ 0 & if x = 1 \end{aligned}$
3. $f (x) = {\begin{aligned} x & if 0 < x \leq 1 \\ 1 & if x = 0 \end{aligned}$ $f (x) = {\begin{aligned} x & if 0 < x \leq 1 \\ 1 & if x = 0 \end{aligned}$
4. $f (x) = {\begin{aligned} 0 & if x = 0 or x = 1 \\ 1 & if 0 < x < 1 and x is rational \\ - 1 & if 0 < x < 1 and x is irrational \end{aligned}$ $f (x) = {\begin{aligned} 0 & if x = 0 or x = 1 \\ 1 & if 0 < x < 1 and x is rational \\ - 1 & if 0 < x < 1 and x is irrational \end{aligned}$

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