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]

Answer :

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.


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