Q. 1 N5.0( 1 Vote )

# Verify Lagrange’s mean value theorem for the following functions on the indicated intervals. In each find a point ‘c’ in the indicated interval as stated by the Lagrange’s mean value theorem :f(x) = x2 + x – 1 on [0, 4]

Lagrange’s mean value theorem states that if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there is at least one point x=c on this interval, such that

f(b)−f(a)=f′(c)(b−a)

This theorem is also known as First Mean Value Theorem.

f(x) = x2 + x – 1 on [0, 4]

Every polynomial function is continuous everywhere on (−∞, ∞) and differentiable for all arguments.

Here, f(x) is a polynomial function. So it is continuous in [0, 4] and differentiable in (0, 4). So both the necessary conditions of Lagrange’s mean value theorem is satisfied.

f(x) = x2 + x – 1

Differentiating with respect to x:

f’(x) = 2x + 1

For f’(c), put the value of x=c in f’(x):

f’(c)= 2c + 1

For f(4), put the value of x=4 in f(x):

f(4)= (4)2 + 4 – 1

= 16 + 4 – 1

= 19

For f(0), put the value of x=0 in f(x):

f(0) = (0)2 + 0 – 1

= 0 + 0 – 1

= – 1

2c + 1 = 5

2c = 5 – 1

2c = 4

Hence, Lagrange’s mean value theorem is verified.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses