1. Tardigrade
  2. Question
  3. Mathematics
  4. Let F: R arrow R be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F prime(x)<0 for all x ∈(1 / 2,3). Let f(x)=x F(x) for all x ∈ R. If ∫ limits13 x2 F prime(x) d x=-12 and ∫ limits13 x3 F prime prime(x) d x=40, then the correct expression(s) is(are)

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $F: R \rightarrow R$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F^{\prime}(x)<0$ for all $x \in(1 / 2,3)$. Let $f(x)=x F(x)$ for all $x \in R$.
If $\int\limits_1^3 x^2 F^{\prime}(x) d x=-12$ and $\int\limits_1^3 x^3 F^{\prime \prime}(x) d x=40$, then the correct expression(s) is(are)

Integrals

Solution:

$\int\limits_1^3 x^3 F^{\prime \prime}(x) d x=40 \left[x^3 F^{\prime}(x)\right]_1^3-\int\limits_1^3 3 x^2 F^{\prime}(x) d x=40$
$\Rightarrow \left[x^2 f^{\prime}(x)-x f(x)\right]_1^3-3(-12)=40$
$\Rightarrow 9 f^{\prime}(3)-3 f(3)-f^{\prime}(1)+f(1)=4$
$\Rightarrow 9 f^{\prime}(3)+36-f^{\prime}(1)+0=4 $
$ \Rightarrow 9 f^{\prime}(3)-f^{\prime}(1)+32=0 \Rightarrow (3)$
$ \Rightarrow \int\limits_1^3 x^2 F^{\prime}(x) d x=-12$
$ \Rightarrow {\left[x^2 F(x)\right]_1^3-\int\limits_1^3 2 x F(x) d x=-12} $
$ \Rightarrow -36-2 \int\limits_1^3 f(x) d x=-12 $
$ \Rightarrow \int\limits_1^3 f(x) d x=-12 \Rightarrow (4)$