Verify Rolle’s theorem for the function f (x) = x 2 − 4x + 4 in the interval[1,3]
Hi!
Here is the answer to your question.
The given function is f (x) = x2 – 4x + 4
f (x) is a polynomial function, so it continuous and differentiable everywhere
(i) f (x) being polynomial function, is continuous on [1, 3]
(ii) f (x) is differentiable on (1, 3)
(iii) f (x) = (1)2 – 4 + 4 = 1
f (3) = (3)2 – 4 × 3 + 4 = 1
∴ f (1) = f (3)
Thus, all the conditions of Rolle’s Theorem are satisfied.
So, there must exist a real number c ε (1, 3) such that f '(c) = 0
Now, f '(c) = 0
⇒ 2c – 4 = 0
⇒ c = 2
Clearly, 2 ε (1, 3)
Thus, c = 2 ε (1, 3) such that f '(c) = 0
Hence, Rolle’s Theorem is verified.
Cheers!