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Mathematics
Mathematics
Problem
If a function
f
(
x
)
f(x)
f
(
x
)
defined by
f
(
x
)
=
{
a
e
x
+
b
e
−
x
,
−
1
≤
x
<
1
c
x
2
,
1
≤
x
≤
3
a
x
2
+
2
c
x
,
3
<
x
≤
4
f(x) = \begin{cases} ae^x + be^{-x}, & -1 \le x < 1 \\ cx^2, & 1 \le x \le 3 \\ ax^2 + 2cx, & 3 < x \le 4 \end{cases}
f
(
x
)
=
⎩
⎨
⎧
a
e
x
+
b
e
−
x
,
c
x
2
,
a
x
2
+
2
c
x
,
−
1
≤
x
<
1
1
≤
x
≤
3
3
<
x
≤
4
be continuous for some
a
,
b
,
c
∈
R
a, b, c \in \mathbb{R}
a
,
b
,
c
∈
R
and
f
′
(
0
)
+
f
′
(
2
)
=
e
f'(0) + f'(2) = e
f
′
(
0
)
+
f
′
(
2
)
=
e
, then the value of
a
a
a
is : [JEE-MAIN ONLINE - 2020] (1)
e
e
2
−
3
e
−
13
\frac{e}{e^2 - 3e - 13}
e
2
−
3
e
−
13
e
(2)
e
e
2
+
3
e
+
13
\frac{e}{e^2 + 3e + 13}
e
2
+
3
e
+
13
e
(3)
1
e
2
−
3
e
+
13
\frac{1}{e^2 - 3e + 13}
e
2
−
3
e
+
13
1
(4)
e
e
2
−
3
e
+
13
\frac{e}{e^2 - 3e + 13}
e
2
−
3
e
+
13
e
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