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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