Question

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=x+4 e^{x} $$

Short answer

The function \(y=x+4e^{x}\) has a horizontal tangent at the point \((-ln4, -ln4+1)\).

Step-by-step solution

Find the derivative of the function

The original function is \(y=x+4e^{x}\). This function is a sum of two functions,\( f(x)=x \) and \( g(x)=4e^{x} \). The derivative of \(x\) respect to \(x\) is 1 and the derivative of \(4e^{x} \) respect to \(x\) is \(4e^{x} \). Thus, the derivative of the original function \( y' \) is \( y'=1+4e^{x}\) .

Set the derivative equal to zero and solve for \(x\)

The points with a horizontal tangent line are those where the derivative is zero. Therefore, \(1+4e^{x}=0\) The solution for \(x\) can be found by isolating the exponential term and then applying the natural logarithm (ln) to both sides. Thus \(x = -ln4\).

Find the corresponding y-value

Hence, \(y=-ln4+4e^{-ln4} = -ln4+1=-ln4+1\). Thus, the graph of the function has a horizontal tangent at the point \((-ln4, -ln4+1)\).