Question

The equation of the tangent to the curve $y = 4x e^x$ at $\left(-1, -\frac{4}{e}\right)$ is

• A. $6x - e^4 y = -5$
• B. $x - e^4 y = 0$
• C. $x = -1$
• D. $y = -\frac{4}{e}$

Hint:

Whenever the slope of a line evaluates to exactly zero ($m = 0$), the line is horizontal. The equation of any horizontal line is simply fixed as $y = c$, where $c$ is the y-coordinate of the point it passes through! This lets you pick option (D) instantly without doing any line configuration setup.

Answer 1

The Correct Option is D

Step 1: Find the slope by differentiating.
For $y=4xe^x$, use the product rule. $\frac{dy}{dx}=4e^x+4xe^x=4e^x(1+x)$.

Step 2: Evaluate the slope at the point.
At $x=-1$, $\frac{dy}{dx}=4e^{-1}(1-1)=0$. The slope is zero, so the tangent is horizontal.

Step 3: Write the tangent line.
A horizontal line through the point $\left(-1,-\frac{4}{e}\right)$ keeps the same $y$ value everywhere. \[ y=-\frac{4}{e} \]
\[ \boxed{y=-\frac{4}{e}} \]