Question

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y' - y''$ at $x = -1$ is equal to

• A. 976
• B. 464
• C. 496
• D. 944

Answer 1

Answer: (C)

496

Answer 2

1. Simplify $y(x)$: Multiply $y(x)$ by $(1-x)$ to get $(1-x)y(x) = 1-x^{32}$.
2. Differentiate implicitly:
   • First derivative: $-y(x) + (1-x)y'(x) = -32x^{31}$.
   • Second derivative: $-2y'(x) + (1-x)y''(x) = -992x^{30}$.
3. Evaluate at $x=-1$:
   • $y(-1) = 0$ (due to the $(1+x)$ factor).
   • Substitute $y(-1)=0$ into the first derivative equation to find $y'(-1) = 16$.
   • Substitute $y'(-1)=16$ into the second derivative equation to find $y''(-1) = -480$.
4. Calculate $y'(-1) - y''(-1)$: $16 - (-480) = 496$.