Verify Rolle's theorem for f(x) = x2 - 5x + 6 in [2,3]

1.	Verify Rolle's theorem for f(x) = x2 - 5x + 6 in [2,3]
Answer» Given, f(x) = x² - 5x + 6 in [2,3] Since, f(x) being a polynomial function in x. So, f(x) in differential and continuous every where. So, f(x) is continuous in [2,3] and derivable is (2,3) Also, f(2) = 0 an f(3) = 0 ∴ f(2) = f(3) Hence, all the condition of Rolle's theorem are satisfied for f(x) in [2,3] So, there exist C ∈ (2,3) such that f'(C) = 0 Here, f(x) = 2x - 5 So, f'(C) = 0 2C - 5 = 0 C = 5/2 ∴ C = 5/2 ∈ (2,3) such that f'(C) = 0 Hence, Rolle's theorem verified.

Answer»

Given, f(x) = x² - 5x + 6 in [2,3]

Since, f(x) being a polynomial function in x.

So, f(x) in differential and continuous every where.

So, f(x) is continuous in [2,3] and derivable is (2,3)

Also, f(2) = 0 an f(3) = 0

∴ f(2) = f(3)

Hence, all the condition of Rolle's theorem are satisfied for f(x) in [2,3]

So, there exist C ∈ (2,3) such that f'(C) = 0

Here, f(x) = 2x - 5

So, f'(C) = 0

2C - 5 = 0

C = 5/2

∴ C = 5/2 ∈ (2,3) such that f'(C) = 0

Hence, Rolle's theorem verified.

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