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Joint Entrance Examination
Mathematics
Application of Derivatives
Question 183
Question 183 of 212
Medium
MCQ
The function \[f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}\], is :
A
increasing in \[\left(-\frac{1}{2}, 1\right)\]
B
decreasing in \[\left(\frac{1}{2}, 2\right)\]
C
increasing in \[\left(-1,-\frac{1}{2}\right)\]
D
decreasing in \[\left(-\frac{1}{2}, \frac{1}{2}\right)\]
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Official solution
\[f(x) = x{e^{x(1 - x)}},\,x \in R\] \[f'(x) = x{e^{x(1 - x)}}\,.\,(1 - 2x) + {e^{x(1 - x)}}\] \[ = {e^{x(1 - x)}}[x - 2{x^2} + 1]\] \[ = - {e^{x(1 - x)}}[2{x^2} - x - 1]\]
=
−
e
x
(
1
−
x
)
(
2
x
+
1
)
(
x
−
1
)
= - {e^{x(1 - x)}}(2x + 1)(x - 1)
=
−
e
x
(
1
−
x
)
(
2
x
+
1
)
(
x
−
1
)
\[\therefore\]
f
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x
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f(x)
f
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x
)
is increasing in \[\left( { - {1 \over 2},1} \right)\] and decreasing in \[\left( { - \infty ,\, - {1 \over 2}} \right) \cup \left( {1,\infty } \right)\]
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