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Joint Entrance Examination
Mathematics
Limits, Continuity and Differentiability
Question 381
Question 381 of 401
Medium
MCQ
Consider the function \[f:(0,2) \rightarrow \mathbf{R}\] defined by \[f(x)=\frac{x}{2}+\frac{2}{x}\] and the function
g
(
x
)
g(x)
g
(
x
)
A
\[g\] is continuous but not differentiable at \[x=1\]
B
\[g\] is continuous and differentiable for all \[x \in(0,2)\]
C
\[g\] is not continuous for all \[x \in(0,2)\]
D
\[g\] is neither continuous nor differentiable at \[x=1\]
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Official solution
\[
f
:
(
0
,
2
)
→
R
;
f
(
x
)
=
x
2
+
2
x
f
′
(
x
)
=
1
2
−
2
x
2
\begin{aligned} & f:(0,2) \rightarrow R ; f(x)=\frac{x}{2}+\frac{2}{x} \\ & f^{\prime}(x)=\frac{1}{2}-\frac{2}{x^2} \end{aligned}
f
:
(
0
,
2
)
→
R
;
f
(
x
)
=
2
x
+
x
2
f
′
(
x
)
=
2
1
−
x
2
2
\] \[\therefore \mathrm{f}(\mathrm{x})\] is decreasing in domain. \[g(x)=
{
x
2
+
2
x
0
<
x
≤
1
3
2
+
x
1
<
x
<
2
\begin{cases}\frac{x}{2}+\frac{2}{x} & 0 < x \leq 1 \\ \frac{3}{2}+x & 1 < x < 2\end{cases}
{
2
x
+
x
2
2
3
+
x
0
<
x
≤
1
1
<
x
<
2
\]
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