Question

If f(x) = (cosx + isinx)(cos3x + isin3x)..........

(cos(2n–1)x + isin(2n – 1)x), then f ′′(x) is equal to

• A. n²f(x)
• B. –n⁴f(x)
• C. –n²f(x)
• D. n⁴f(x)

Answer 1

Answer: (B)

–n⁴f(x)

Answer 2

We have,

f(x) = cos(x + 3x +...+(2n – 1)x) + i sin(x + 3x + 5x ... +

(2n – 1)x)

= cos n²x + i sin n²x

⇒ f ′(x) = –n²(sin n²x) + n²(i cos n²x)

⇒ f ′′(x) = –n⁴ cos n²x – n⁴i sin n²x

⇒ f ′′(x) = –n⁴(cos n²x + i sin n²x)

⇒ f ′′(x) = –n⁴f(x)