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Mathematics
Mathematics
Problem
Let
f
:
[
1
,
3
]
→
[
0
,
∞
)
f:[1,3]\rightarrow[0,\infty)
f
:
[
1
,
3
]
→
[
0
,
∞
)
be continuous and differentiable function and if
(
f
(
3
)
−
f
(
1
)
)
⋅
(
f
2
(
3
)
+
f
2
(
1
)
+
f
(
3
)
f
(
1
)
)
=
k
f
2
(
c
)
f
′
(
c
)
(f(3)-f(1))\cdot(f^2(3)+f^2(1)+f(3)f(1))=kf^2(c)f'(c)
(
f
(
3
)
−
f
(
1
))
⋅
(
f
2
(
3
)
+
f
2
(
1
)
+
f
(
3
)
f
(
1
))
=
k
f
2
(
c
)
f
′
(
c
)
where
c
∈
(
1
,
3
)
c\in(1,3)
c
∈
(
1
,
3
)
, then find the value of
k
k
k
.
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