Question

Find c of Lagrange's mean value theorem f(x)=x(x-1)(x-2) where x€(0,1/2)

Answer 1

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Answer 2

The value of c is (6±√21)/6.

Given,

A function f(x) = x(x-1)(x-2) where x€(0,1/2)

To Find,

The c of Lagrange's mean value theorem.

Solution,

The given function is

f(x) = x(x-1)(x-2) where x€(0,1/2)

f(x) = x³-3x²+2x

Now, according to Lagrange's mean value theorem there exists some c€(0,1/2) such that f'(c) = f(1/2)-f(0)/1/2-0

f(1/2) = 3/8

f(0) = 0

f'(x) = 3x²-6x+2

Now,

3x²-6x+2 = (3/8-0)/(1/2-0)

3x²-6x+2 = 3/4

12x²-24x+5 = 0

Using the quadratic formula

x = (6±√21)/6 = c

Hence, the value of c is (6±√21)/6.

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