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.