Consider the function $f:(0,2) \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0

Question

Consider the function $f:(0,2) \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by

$$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x < 2 \end{array} .\right. \text { Then, }$$

$g$ is continuous but not differentiable at $x=1$

$g$ is continuous and differentiable for all $x \in(0,2)$

$g$ is not continuous for all $x \in(0,2)$

$g$ is neither continuous nor differentiable at $x=1$

Solution

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