multiple choice 1 point\nthe function f has a continuous derivative. the table above gives values of f and…

Answer:

D. -7

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int_{a}^{b}u\mathrm{d}v=uv|{a}^{b}-\int{a}^{b}v\mathrm{d}u$. Let $u = x$ and $\mathrm{d}v=f^{\prime}(x)\mathrm{d}x$. Then $\mathrm{d}u=\mathrm{d}x$ and $v = f(x)$. So, $\int_{0}^{4}xf^{\prime}(x)\mathrm{d}x=xf(x)|{0}^{4}-\int{0}^{4}f(x)\mathrm{d}x$.

Step2: Evaluate $xf(x)|_{0}^{4}$

$xf(x)|_{0}^{4}=4f(4)-0\times f(0)$. From the table, $f(4)= - 3$ and $f(0)=2$. So, $4f(4)-0\times f(0)=4\times(-3)-0=-12$.

Step3: Substitute the value of $\int_{0}^{4}f(x)\mathrm{d}x$

We know that $\int_{0}^{4}f(x)\mathrm{d}x = 8$.

Step4: Calculate $\int_{0}^{4}xf^{\prime}(x)\mathrm{d}x$

$\int_{0}^{4}xf^{\prime}(x)\mathrm{d}x=xf(x)|{0}^{4}-\int{0}^{4}f(x)\mathrm{d}x=-12 - 8=-7$.