No Problemmo #4

Calculus Level 4

Let \(F : \mathbb R \to \mathbb R \) be a thrice differentiable function. Suppose that \(F(1) = 0, F(3) =-4\) and \(F'(x) < 0 \) for all \(x \in \left( \frac12, 3 \right) \).

Let \(f(x) = x F(x) \) for all \(x\in \mathbb R\).

If \(\displaystyle \int_1^3 x^2 F'(x) \, dx = -12\) and \(\int_1^3 x^3 F''(x) \, dx = 40 \), then which of the following statements are true?

  1. \(f'(1)<0\)
  2. \(f(2)<0\)
  3. \(f'(x)\neq 0\) for any \(x \in (1,3)\)
  4. \(f'(x)=0\) for some \(x \in (1,3)\)
  5. \(9f'(3)+f'(1)-32 = 0\)
  6. \(\int _{ 1 }^{ 3 }{ f(x)dx } = 12\)
  7. \(9f'(3)-f'(1)+32 = 0\)
  8. \(\int _{ 1 }^{ 3 }{ f(x)dx } = -12\)

Enter your answer in the increasing sequence of numbers. For eg. If options 2,4 and 8 are correct, then input 248 as the answer.

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