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 :
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.
Here,
⇒ x2 – 4 >0
⇒ x2 > 4
⇒ f(x) exists for all values expect ( – 2, 2)
∴ f(x) is continuous in [2, 4]
Differentiating with respect to x:
Here also,
⇒ f’(x) exists for all values of x except (2, – 2)
∴ f(x) is differentiable in (2, 4)
So both the necessary conditions of Lagrange’s mean value theorem is satisfied.
On differentiating with respect to x:
For f’(c), put the value of x=c in f’(x):
For f(4), put the value of x=4 in f(x):
For f(2), put the value of x=2 in f(x):
⇒ f(2) = 0
Squaring both sides:
⇒ c2 = 3(c2 – 4)
⇒ c2 = 3c2 – 12
⇒ – 2c2 = – 12
⇒ c2 = 6
Hence, Lagrange’s mean value theorem is verified.