The function sinx (1 + cosx) at x =π/3 isa)Maximab)Minimac)Both (A) & (B)d)None of theseCorrect answer is option 'A'. Can you explain this answer?

Question

The function sinx (1 + cosx) at x =&pi;/3 isa)Maximab)Minimac)Both (A) & (B)d)None of theseCorrect answer is option 'A'. Can you explain this answer?

Answer

Let f(x) = Sin x ( 1 + cos x)

⇒ f'(x) = cosx (1 + cosx) + sinx (–sinx) = cosx + cos²x – sin²x

= cosx + cos²x

f"(x) = – sin x – 2sin2x = – (sinx + 2sin2x)

for maximum or minimum value of f(x), f'(x) = 0

therefore cos x + cos2 x = 0

⇒ cos x = – cos2x

⇒ cosx = –cos (π ± 2x)

⇒ x = π ± 2x

⇒ x = ��3 , –π

Now

Therefore f(x) is maximum at x =π/3.