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Mathematics
Mathematics
Problem
Let the average value of a continuous function
f
f
f
on
[
a
,
b
]
[a,b]
[
a
,
b
]
be
M
M
M
. For constants
A
,
B
(
>
0
)
,
C
,
D
A,B(>0),C,D
A
,
B
(
>
0
)
,
C
,
D
, consider
g
(
x
)
=
A
f
(
B
x
+
C
)
+
D
g(x)=A f(Bx + C)+D
g
(
x
)
=
A
f
(
B
x
+
C
)
+
D
on the interval
[
a
−
C
B
,
b
−
C
B
]
\bigl[\dfrac{a-C}{B},\,\dfrac{b-C}{B}\bigr]
[
B
a
−
C
,
B
b
−
C
]
. Express the average value of
g
g
g
on this interval in terms of
A
,
D
,
A, D,
A
,
D
,
and
M
M
M
.
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