Question

Verify Lagrange's Mean value theorem Where Function :f(x)=(x-1) (x-2)(x-3) is in the interval [0,4]. ​

Answer 1

Given, f(x)=

x

2

−x

,x∈[1,4]

Since x

2

−x is continuous on R

and

x

2

−x

is exists in [1,4]

⇒f(x) is continuous in [1,4].

Differentiating the given function w.r.t. x

f

′

(x)=

2

1

(x

2

−x)

−1/2

.(2x−1)=

2

x

2

−x

2x−1

which exists ∀x∈R.

∴f(x) is differentiable in (1,4).

Thus, both the conditions of Lagrange's mean value theorem is satisfied therefore, ∃c in (1,4).

Such that f

′

(c)=

4−1

f(4)−f(1)

2

c

2

−c

2c−1

=

3

12

3(2c−1)=2

c

2

−c

.

12

9(4c

2

−4c+1)=48(c

2

−c)

3(4c

2

−4c+1)=16(c

2

−c)

⇒12c

2

−12c+3=16c

2

−16c

⇒4c

2

−4c−3=0

⇒(2c−3)(2c+1)=0

c=

2

3

,

2

−1

Thus, ∃ (there exist) c=

2

3

∈(1,4)

Such that f

′

(

2

3

)=

4−1

f(4)−f(1)

Hence, Lagrange's mean value theorem is verified and c=

2

3

.