3(sin x − cos x) + 4(cos3 x − sin3 x) का अधिकतम मान क्या है ?

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NDA 02/2022 Mathematics Official Paper (Held On 04 Sep 2022)
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  1. 1
  2. \(\sqrt{2}\)
  3. \(\sqrt{3}\)
  4. 2

Answer (Detailed Solution Below)

Option 2 : \(\sqrt{2}\)
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संकल्पना:

  • sin3x = 3sinx - 4sin3x
  • cos3x = 4cos3x - 3cosx

गणना:

f(x) = 3(sin x − cos x) + 4(cos3 x − sin3 x)

f(x) = 3sinx - 3cosx + 4cos3x - 4sin3x

f(x) = (3sinx - 4sin3x) + (4cos3x - 3cosx)

f(x) = sin3x + cos3x

f'(x) = 3cos3x - 3sin3x

अब, f'(x) = 0

⇒  3cos3x - 3sin3x = 0

⇒ cos3x - sin3x = 0

⇒ sin3x = cos3x

⇒ tan3x = 1

⇒ \(3x = \frac{\pi }{4}\)

अत: दिए गए फलन का अधिकतम मान है = \(sin\frac{\pi }{4}+cos\frac{\pi }{4}\)

\(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}\)

  सही विकल्प (2) है 

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