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Question
Verify Rolle’s Theorem for the function f(x) = ex (sin x – cos x) on `[ (π)/(4), (5π)/(4)]`.
Solution
f (x) = ex (sin x - cos x), `[ (π)/(4), (5π)/(4)]`
Differentiate w.r.t. x, we have
f (x) = ex (cos x + sin x) + ( sin x - cos x) ex
f (x) = ex [cos x + sin x + sin x - cos x]
f (x) = 2 ex sin x
For maxima or minima, we have
f'(x) = 0
2ex sin x = 0
sin x = 0
x = n π
Thus, x = π, ∵ lies between `[ (π)/(4), (5π)/(4)]`
So, Rolle's theorem is verified.
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