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Mathematics
Mathematics
Problem
Consider the function
f
(
x
)
=
k
sin
x
+
m
x
f(x)=k\sin x+m x
f
(
x
)
=
k
sin
x
+
m
x
on the interval
[
−
π
2
,
π
2
]
\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]
[
−
2
π
,
2
π
]
, where
k
k
k
and
m
m
m
are constants. Given that
f
(
0
)
=
0
,
(
f
−
1
)
′
(
0
)
=
1
4
,
and
f
(
π
2
)
=
3
,
f(0)=0,\quad (f^{-1})'(0)=\dfrac{1}{4},\quad \text{and}\quad f\!\left(\dfrac{\pi}{2}\right)=3,
f
(
0
)
=
0
,
(
f
−
1
)
′
(
0
)
=
4
1
,
and
f
(
2
π
)
=
3
,
determine the values of
k
k
k
and
m
m
m
.
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