\(f(x) = \frac{1}{\tan x+\cot x},\) ప్రమేయాల యొక్క గరిష్ట విలువ ఎంత, ఇక్కడ  \(0 < x < \frac{\pi}{2}?\)

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  1. \(\frac{1}{4}\)
  2. \(\frac{1}{2}\)
  3. 1
  4. 2

Answer (Detailed Solution Below)

Option 2 : \(\frac{1}{2}\)
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ఉపయోగించిన సూత్రం:

  • sin θ/cos θ = tan θ 
  • cos θ/sin θ = cot θ 
  • sin2θ + cos2θ = 1
  • 2sin θ cos θ = sin 2θ

 

గణన:

\(f(x) = \frac{1}{\tan x+\cot x},\)    \(0 < x < \frac{\pi}{2}?\)

⇒ \(f(x) = \frac{1}{\tan x+\cot x},\)

⇒ \(f(x) = \frac{1}{\frac{sin x}{cosx}+\frac{cosx}{sinx}}\)

⇒ \(f(x) =\frac{sinx\ cosx}{sin^2x+cos^2x}\)

⇒ f(x) = sin x cos x (\(\frac{2}{2}\))    [∵ sin2θ + cos2θ = 1]  

⇒ f(x) = \(\frac{1}{2}\)sin 2x        [∵ 2sin θ cosθ = sin 2θ]

మనకు తెలుసు, -1 ≤ sin θ ≤ 1 

⇒  -1 ≤ sin 2x ≤ 1 

∴  f(x)max = \(\frac{1}{2}\)(1) = 1/2

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