Math, asked by heyshivani03, 5 months ago

If f(x) = 2,g(x) = cos x, then find the point c belongs to0,pi/2 [ which gives the result
of Cauchy's mean value theorem in the interval [0,pi/2] for the functions) fxand
gx.​

Answers

Answered by TheBrainlyKing1
1

Step-by-step explanation:

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = e^2x, [0, 3] No, is continuous on [0, 3] but not differentiate on (0, 3). Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem.

Answered by varshakumari452
1

Answer:

Let the functions

f

(

x

)

and

g

(

x

)

be continuous on an interval

[

a

,

b

]

,

differentiable on

(

a

,

b

)

,

and

g

(

x

)

0

for all

x

(

a

,

b

)

.

Then there is a point

x

=

c

in this interval such that

f

(

b

)

f

(

a

)

g

(

b

)

g

(

a

)

=

f

(

c

)

g

(

c

)

.

Proof.

First of all, we note that the denominator in the left side of the Cauchy formula is not zero:

g

(

b

)

g

(

a

)

0.

Indeed, if

g

(

b

)

=

g

(

a

)

,

then by Rolle’s theorem, there is a point

d

(

a

,

b

)

,

in which

g

(

d

)

=

0.

This, however, contradicts the hypothesis that

g

(

x

)

0

for all

x

(

a

,

b

)

.

We introduce the auxiliary function

F

(

x

)

=

f

(

x

)

+

λ

g

(

x

)

and choose

λ

in such a way to satisfy the condition

F

(

a

)

=

F

(

b

)

.

In this case we get

f

(

a

)

+

λ

g

(

a

)

=

f

(

b

)

+

λ

g

(

b

)

,

f

(

b

)

f

(

a

)

=

λ

[

g

(

a

)

g

(

b

)

]

,

λ

=

f

(

b

)

f

(

a

)

g

(

b

)

g

(

a

)

.

and the function

F

(

x

)

takes the form

F

(

x

)

=

f

(

x

)

f

(

b

)

f

(

a

)

g

(

b

)

g

(

a

)

g

(

x

)

.

This function is continuous on the closed interval

[

a

,

b

]

,

differentiable on the open interval

(

a

,

b

)

and takes equal values at the boundaries of the interval at the chosen value of

λ

.

Then by Rolle’s theorem, there exists a point

c

in the interval

(

a

,

b

)

such that

F

(

c

)

=

0.

Hence,

f

(

c

)

f

(

b

)

f

(

a

)

g

(

b

)

g

(

a

)

g

(

c

)

=

0

or

f

(

b

)

f

(

a

)

g

(

b

)

g

(

a

)

=

f

(

c

)

g

(

c

)

.

By setting

g

(

x

)

=

x

in the Cauchy formula, we can obtain the Lagrange formula:

f

(

b

)

f

(

a

)

b

a

=

f

(

c

)

.

Step-by-step explanation:

hope it helps

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