Final answer:
To find the horizontal tangent line for y = -4x + e^x, the derivative is set to zero resulting in x = ln(4). Substituting this back into the original function, we get y = 0. Therefore, the horizontal tangent line is y = 0 at the point (ln(4), 0).
Step-by-step explanation:
To find the horizontal tangent line for the function y = -4x + e^x, we need to determine where the slope of the tangent to the curve is zero. The slope of the tangent line at any point on the curve is given by the derivative of the function. Let's calculate this step by step.
First, take the derivative of y with respect to x:
y' = -4 + e^x
A horizontal tangent line occurs where y' = 0. Thus, we set the derivative equal to zero and solve for x:
-4 + e^x = 0
Add 4 to both sides:
e^x = 4
Take the natural logarithm of both sides to solve for x:
x = ln(4)
Now, plug x = ln(4) into the original equation to find the y-coordinate of the tangent:
y = -4ln(4) + e^ln(4)
Since e^ln(4) = 4, the equation simplifies to:
y = 0
Therefore, the horizontal tangent line is y = 0 at the point (ln(4), 0).