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Question
Verify Rolle’s theorem for the following function:
f (x) = x2 - 4x + 10 on [0, 4]
Solution
Since f (x) is a polynomial,
(i) It is continuous on [0, 4]
(ii) It is differentiable on (0, 4)
(iii) f (0) = 10, f (4) = 16 - 16 + 10 = 10
∴f(0) = f(4) = 10
Thus all the conditions on Rolle’s theorem are satisfied
The derivative of f (x) should vanish for at least one point c in (0, 4). To obtain the value of c, we
proceed as follows
f(x) = x2 - 4x + 10
f'(x) = 2x - 4 = 2(x - 2)
∴ f'(x) = 0 ⇒ (x - 2) = 0
∴ x= 2
∴ ∃c = 2 in (0,4)
We know that 2 ∈ (0, 4)
Thus Rolle’s theorem is verified.
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