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Question: The function \(f(x) = \cos x - 2px\) is monotonically decreasing for...

The function f(x)=cos⁡x−2pxf(x) = \cos x - 2px is monotonically decreasing for

A

p<12p < \frac{1}{2}

B

p>12p > \frac{1}{2}

C

p<2p < 2

D

p>2p > 2

Answer

p>12p > \frac{1}{2}

Explanation

Solution

f(x)f(x) will be monotonically decreasing, if f′(x)<0f^{'}(x) < 0.

⇒ f′(x)=−sin⁡x−2p<0f^{'}(x) = - \sin x - 2p < 0 ⇒ 12sin⁡x+p>0\frac{1}{2}\sin x + p > 0 ⇒ p>12p > \frac{1}{2}

[∵−1≤sin⁡x≤1]\lbrack\because - 1 \leq \sin x \leq 1\rbrack