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

Question

Question

This question was previously asked in

NDA 02/2022 Mathematics Official Paper (Held On 04 Sep 2022)

Answer 1

संकल्पना:

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