The function f whose graph passes through (4,-20) and whose derivative is cos⁡(4−x) is given by

# The function $f$ whose graph passes through $\left(4,-20\right)$ and whose derivative is $\mathrm{cos}\left(\sqrt{4-x}\right)$ is given by

1. A

$\sqrt{4-x}\mathrm{sin}\sqrt{4-x}+\left(4-6x\right)\mathrm{cos}\sqrt{4-x}$

2. B

$\sqrt{4-x}\mathrm{sin}\sqrt{4-x}+\sqrt{4-x}\mathrm{cos}\sqrt{4-x}-20$

3. C

$-2\left(\sqrt{4-x}\mathrm{sin}\sqrt{4-x}+\mathrm{cos}\sqrt{4-x}\right)-18$

4. D

$-\left(2x+12\right)\sqrt{4-x}\mathrm{sin}\sqrt{4-x}$

$+\left(4-6x\right)\mathrm{cos}\sqrt{4-x}$

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### Solution:

$\begin{array}{r}f\left(x\right)=\int \mathrm{cos}\sqrt{4-x}dx+C\\ =-2\int t\mathrm{cos}tdt,t=\sqrt{4-x}\end{array}$

$\begin{array}{l}=-2\left(\sqrt{4-x}\mathrm{sin}\sqrt{4-x}+\mathrm{cos}\sqrt{4-x}\right)+C\\ -20=f\left(4\right)=-2+C⇒C=-18\end{array}$

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