Mathematics

Mathematics

Problem

Let F:R→RF: \mathbb{R} \to \mathbb{R} be a thrice differentiable function. Suppose that F(1)=0,F(3)=−4F(1) = 0, F(3) = -4 and F′(x)<0F'(x) < 0 for all x∈(1,3)x \in (1, 3). Let f(x)=xF(x)f(x) = xF(x) for all x∈Rx \in \mathbb{R}. If ∫13x2F′(x) dx=−12\int_1^3 x^2 F'(x)\,dx = -12 and ∫13x3F′′(x) dx=40\int_1^3 x^3 F''(x)\,dx = 40, then the correct expression(s) is/are a. 9f′(3)+f′(1)−32=09f'(3) + f'(1) - 32 = 0 b. ∫13f(x)dx=12\int_1^3 f(x) dx = 12 c. 9f′(3)−f′(1)+32=09f'(3) - f'(1) + 32 = 0 d. ∫13f(x)dx=−12\int_1^3 f(x) dx = -12

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