Question

The value of 'c of Lagranges Mean value
Theorem for f(x) = ^x when a=1, b = 4​

Answer 1

Answer:

Step-by-step explanation:

f(x)=

x

​

, where a=1,b=4

Checking conditions for Mean value theorem.

Condition 1:

f(x)=

x

​

 is continuous at (1,4)

Since f(x) is polynomial.

It is continuous in (1,4)

Condition 2:

If f(x) is differentiable

f(x)=

x

​

f(x) is a polynomial and every polynomial function is differentiable

⇒  f(x) is differentiable at x∈[1,4]

Condition 3:

f(x)=

x

​

f

′

(x)=

2

1

​

 

x

​

1

​

f

′

(c)=

2

c

​

1

​

f(a)=f(1)

        =

1

​

=1

f(b)=f(4)

        =

4

​

        =2

By Lagranges mean value theorem,

f

′

(c)=

b−a

f(b)−f(a)

​

2

c

​

1

​

=

4−1

2−1

​

2

c

​

1

​

=

3

1

​

c

​

=

2

3

​

∴  c=

4

9

​