Q. 1 K4.0( 2 Votes )

# Verify Lagrange’s

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) is continuous in [1, 3] Differentiating with respect to x:   Here, f’(x) exists for all values except 0

f(x) is differentiable in (1, 3)

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(3), put the value of x=3 in f(x):   For f(1), put the value of x=1 in f(x): f(1) = 2     6(c2 – 1) = 4c2

6c2 – 6 = 4c2

6c2 – 4c2 = 6

2c2 = 6 c2 = 3 Hence, Lagrange’s mean value theorem is verified.

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