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Question
Verify Rolle’s theorem for the following functions : f(x) = sin x + cos x + 7, x ∈ [0, 2π]
Solution
The functions sin x, cos x and 7 are continuous and differentiable on their domains.
∴ f(x) = sin x + cos x + 7 is continuous on [0, 2π] and differentiable on (0, 2π)
Now, f(0)
= sin 0 + cos 0 + 7
= 0 + 1 + 7 = 8
and f(2π)
= sin π + cos π + 7
= 0 + 1 + 7 = 8
∴ f(0) = f(2π)
Thus, the function f satisfies all the conditions of Rolle's theorem.
∴ there exists c∈ (0, 2π) such that f'(c) = 0.
Now , f(x) = sin x + cos x + 7
∴ f'(x) = `d/dx(sin x + cos x + 7)`
= cos x – sin x + 0
= cos x – sin x
∴ f'(c) = cos c – ic
∴ f'(c) = 0 gives, cos c – sin c = 0
∴ cos c = sin c
∴ c = `pi/(4), (5pi)/(4),(9pi)/(4)`, ...
But `pi/(4), (5pi)/(4) ∈ (0, 2pi)`
∴ c = `pi/(4) or (5pi)/(4)`
Hence, the Rolle's theorem is verified.
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