let $g(x)=x^{2}+2x$; what is the equation that is parallel to normal line of the function thats given at the…

Explanation:

Step1: Find the derivative of $G(x)$

Differentiate $G(x)=x^{2}+2x$ using the power - rule. The derivative $G^\prime(x)=2x + 2$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $G^\prime(x)$. $G^\prime(2)=2\times2+2=6$. This is the slope of the tangent line at the point $(2,8)$.

Step3: Find the slope of the normal line

The slope of the normal line $m_n$ is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line $m_t = 6$, then $m_n=-\frac{1}{6}$.

Step4: Determine the equation of the parallel line

Parallel lines have the same slope. We check the slopes of the given lines. Rewrite the lines in slope - intercept form $y=mx + b$. For $x + 6y=50$, we solve for $y$: $6y=-x + 50$, so $y=-\frac{1}{6}x+\frac{50}{6}$. For $x + 6y=2$, we solve for $y$: $6y=-x + 2$, so $y=-\frac{1}{6}x+\frac{1}{3}$. The lines $y = 6x-4$ and $y = 6x - 12$ have slope $m = 6$. The lines $x + 6y=50$ and $x + 6y=2$ have slope $m=-\frac{1}{6}$.

Answer:

C. $x + 6y=50$, D. $x + 6y=2$