If fx) = 3 sin 2x, is continuous over interval [0, 7) and differentiable over interval (0, 7) then by Rolle's theorem the value of c is...
Answers
Answered by
3
Step-by-step explanation:
The Rolles theorem says that if:
y
=
f
(
x
)
is a continue function in a set
[
a
,
b
]
;
y
=
f
(
x
)
is a derivable function in a set
(
a
,
b
)
;
f
(
a
)
=
f
(
b
)
;
then at least one
c
∈
(
a
,
b
)
as if
f
'
(
c
)
=
0
exists.
So:
y
=
3
sin
(
2
x
)
is a function that is continue in all
R
, and so it is in
[
0
,
2
π
]
;
y
'
=
6
cos
(
2
x
)
is a function continue in all
R
, so our function is derivable in all
R
, so it is in
[
0
,
2
π
]
;
f
(
0
)
=
f
(
2
π
)
=
0
.
To find
c
, we have to solve:
y
'
(
c
)
=
0
⇒
6
cos
(
2
c
)
=
0
⇒
cos
(
2
c
)
=
0
⇒
2
c
=
π
2
+
2
k
π
⇒
c
=
π
4
+
k
π
⇒
c
1
=
π
4
and
c
2
=
π
4
+
π
=
5
4
π
.
(both the values are
∈
[
0
,
2
π
]
).
Answer link
Similar questions