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Question
Verify Rolle’s theorem for the following functions : f(x) = x2 – 5x + 9, x ∈ 1, 4].
Solution
The functon f gives as f(x) = x2 – 5x + 9 s a polynomial function. Hence it is continuous on [1, 4] and differentiable on (1, 4).
Now, f(1)
= 12 – 5(1) + 9
= 1 – 5 + 9
= 5
and f(4)
= 42 – 5(4) + 9
= 16 – 20 + 9
= 5
∴ f(1) = f(4)
Thus, the function f satisfies all the conditions of the Rolle's theorem.
∴ there exists c ∈ (1, 4) such that f'(c) = 0.
Now, f(x) = x2 – 5x + 9
∴ f'(x) = `d/dx(x^2 - 5x +9)`
= 2x – 5 x 1 + 0
= 2x – 5
∴ f'(c) = 2c – 5
∴ f'(c) = 0 gives, 2c – 5 = 0
∴ c = `(5)/(2) ∈(1, 4)`
Hence, the Rolle's theorem is verified.
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