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Let M be the maximum value of the function f(x) = sin x·(1 + cos x)…
Question
Let M be the maximum value of the function f(x) = sin x·(1 + cos x) over all real x. Compute ⌊100·M⌋ (the floor of one hundred times M).
✓ Verified answer: 129checked by our engine — not a guess
Step-by-step solution
Differentiate f(x) = sin x + sin x cos x = sin x + (1/2) sin 2x. f'(x) = cos x + cos 2x = cos x + 2cos²x − 1 = (2 cos x − 1)(cos x + 1).
f'(x) = 0 when cos x = 1/2 or cos x = −1. At cos x = −1, sin x = 0 and f = 0. At cos x = 1/2, sin x = ±√3/2; the maximum uses sin x = √3/2:
f = (√3/2)(1 + 1/2) = (√3/2)(3/2) = 3√3/4 ≈ 1.29904.
So M = 3√3/4 ≈ 1.29904, and 100·M ≈ 129.904, hence ⌊100·M⌋ = 129.
Final answer129
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