_{f(x) = x (x – 2)}

_{f(x) = x}² – 2x

Since, a polynomial function is always continuous and differentiable.

As f(x) is a polynomial function so it is always continuous on 1, 2 and differentiable on 1, 2.

f(x) satisfies both the conditions of Lagrange’s Theorem on 1, 2.

So, a real number has to exist c Є 1, 2, such that

f(x) = x² – 2x

f'(x) = 2x – 2

f(1) = 1² – 2(1)

= 1 – 2

= -1

f(2) = 2² – 2(2)

= 4 – 4

= 0

= 1

So, 2x – 2 = 1

2x = 3

Є (1, 2)

Hence, Option (D) is the answer.