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Question
Verify Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.
Solution
Consider f(x) = sin 2x in `[0, pi/2]`
Note that:
(i) The function f is continuous in `[0, pi/2]`, as f is a sine function, which is always continuous.
(ii) f′(x) = 2cos 2x, exists in `(0, pi/2)`, hence f is derivable in `(0, pi/2)`.
(iii) f(0) = sin0 = 0 and `"f"(pi/2)` = sinπ = 0
⇒ f(0) = `"f"(pi/2)`.
Conditions of Rolle’s theorem are satisfied. Hence there exists at least one c ∈ `(0, pi/2)` such that f′(c) = 0.
Thus 2 cos 2c = 0
⇒ 2c = `pi/2`
⇒ c = `pi/4`.
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