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) = tan – 1 x on [0, 1]
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) = tan – 1 x on [0, 1]
tan – 1 x has unique value for all x between 0 and 1.
∴ f(x) is continuous in [0, 1]
f(x) = tan – 1 x
Differentiating with respect to x:
x2 always has value greater than 0.
⇒ 1 + x2 > 0
∴ f(x) is differentiable in (0, 1)
So both the necessary conditions of Lagrange’s mean value theorem is satisfied.
⇒ f’(c)= f(1) – f(0)
f(x) = tan – 1 x
Differentiating with respect to x:
For f’(c), put the value of x=c in f’(x):
For f(1), put the value of x=1 in f(x):
f(1) = tan – 1 1
For f(0), put the value of x=0 in f(x):
f(0) = tan – 1 0
⇒ f(0) = 0
f’(c)= f(1) – f(0)
Hence, Lagrange’s mean value theorem is verified.