Question

If f(x)= 3Sin2x, is continuous over the interval [0, 1], differential over (0, x) then by Rolles theorem the value of "C" is

Answer 1

Answer:

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

]

).

Step-by-step explanation: