Consider the function f:(0,2) R defined by f(x)=x 2 +2 x and the … - Vidyadip

MCQ

Consider the function $\mathrm{f}:(0,2) \rightarrow \mathrm{R}$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function$ g(x)$ defined by $g(x)=\left\{\begin{array}{cc}\min \{f(t)\}, & 0 < t \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x< 2\end{array}\right.$. Then

✓
$g$ is continuous but not differentiable at $x=1$
B
$\mathrm{g}$ is not continuous for all $\mathrm{x} \in(0,2)$
C
$g$ is neither continuous nor differentiable at $x=1$
D
$g$ is contimuous and differentiable for all $\mathrm{x} \in(0,2)$

✓

Answer

Correct option: A.

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

a
$ f:(0,2) \rightarrow R ; f(x)=\frac{x}{2}+\frac{2}{x} $

$ f^{\prime}(x)=\frac{1}{2}-\frac{2}{x^2}$

$\therefore \mathrm{f}(\mathrm{x})$ is decreasing in domain.

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