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Question
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
Solution
Given that: f(x) = `sqrt(3)` sinx – cosx – 2ax + b, a ≥ 1
Differentiating both sides w.r.t. x, we get
f'(x) = `sqrt(3) cos x + sin x - 2"a"`
For decreasing function, f'(x) < 0
∴ `sqrt(3) cos x + sin x - 2"a" < 0`
⇒ `2(sqrt(3)/2 cos x + 1/2 sin x) - 2"a" < 0`
⇒ `sqrt(3)/2 cos x + 1/2 sin x - "a" < 0`
⇒ `(cos pi/6 cos x + sin pi/6 sin x) - "a" < 0`
⇒ `cos(x - pi/6) - "a " < 0`
Since cos x ∈ [– 1, 1] and a ≥ 1
∴ f'(x) < 0
Hence, the given function is decreasing in R.
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