Q. 1 G5.0( 1 Vote )

# 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) = 2x – x2 on [0, 1]

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, 1] and 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) = 2x – x2

Differentiating with respect to x:

f’(x) = 2 – 2x

For f’(c), put the value of x=c in f’(x):

f’(c)= 2 – 2c

For f(1), put the value of x=1 in f(x):

f(1)= 2(1) – (1)2

= 2 – 1

= 1

For f(0), put the value of x=0 in f(x):

f(0) = 2(0) – (0)2

= 0 – 0

= 0

f’(c) = f(1) – f(0)

2 – 2c = 1 – 0

– 2c = 1 – 2

– 2c = – 1 Hence, Lagrange’s mean value theorem is verified.

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