Write the equation of the line tangent to the graph of $y = 2 x e ^ { - x } + 6$ at $x = 1$
A) $y = 2 e ^ { - 1 } x + 6$
B) $y = 2 e ^ { - 1 } x - 6$
C) $y = 2 e ^ { - 1 } x$
D) $y = 2 e ^ { - 1 } + 6$
E) $y = 2 e ^ { - 1 } - 6$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line at $x = 1$, we first need to find the derivative of $y = 2xe^{-x} + 6$, which gives us $y' = 2e^{-x} - 2xe^{-x}$. Evaluating this derivative at $x = 1$ gives $2e^{-1} - 2e^{-1} = 0$. The slope of the tangent line at $x = 1$ is 0, indicating a horizontal line. The y-value of the function at $x = 1$ is $2e^{-1} + 6$. Therefore, the equation of the tangent line is $y = 2e^{-1} + 6$.

Write the Equation of the Line Tangent to the Graph y=2xe−x+6y = 2 x e ^ { - x } + 6y=2xe−x+6

Write the Equation of the Line Tangent to the Graph $y = 2 x e ^ { - x } + 6$