Given as the function is f (x) = x2 – 4x + 3 on [1, 3]
Thus, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R. Let us find the values at extremes:
⇒ f (1) = 12 – 4(1) + 3
⇒ f (1) = 1 – 4 + 3
⇒ f (1) = 0
⇒ f (3) = 32 – 4(3) + 3
⇒ f (3) = 9 – 12 + 3
⇒ f (3) = 0
∴ f (1) = f(3), Rolle’* theorem applicable for function ‘f’ on [1,3].
Let us find the derivative of f(x)
⇒ f’(x) = 2x – 4
f’(c) = 0, c ϵ (1, 3), from the definition of Rolle’* Theorem.
⇒ f’(c) = 0
⇒ 2c – 4 = 0
⇒ 2c = 4
⇒ c = 4/2
⇒ C = 2 ϵ (1, 3)
Thus, Rolle’* Theorem is verified.
