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Joint Entrance Examination
Mathematics
Application of Derivatives
Question 61
Question 61 of 212
Medium
NUMERICAL
Let
f
(
x
)
=
2
x
−
x
2
,
x
∈
R
f(x)={2}^{x}-{x}^{2},x\in R
f
(
x
)
=
2
x
−
x
2
,
x
∈
R
. If
m
m
m
and
n
n
n
are respectively the number of points at which the curves
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
and
y
=
f
′
(
x
)
y={f}^{'}(x)
y
=
f
′
(
x
)
intersect the
x
x
x
-axis, then the value of
m
+
n
m+n
m
+
n
is \_\_\_\_\_\_\_\_\_\_\_. Answer: 5
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Official solution
∴
m
=
3
f
′
(
x
)
=
2
x
ln
2
−
2
x
=
0
2
x
ln
2
=
2
x
\begin{gathered} \therefore m=3 \\ {f}^{'}(x)={2}^{x}\ln 2-2x=0 \\ {2}^{x}\ln 2=2x \end{gathered}
∴
m
=
3
f
′
(
x
)
=
2
x
ln
2
−
2
x
=
0
2
x
ln
2
=
2
x
∴
n
=
2
⇒
m
+
n
=
5
\begin{gathered} \therefore n=2 \\ \Rightarrow m+n=5 \end{gathered}
∴
n
=
2
⇒
m
+
n
=
5
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