Graph the derivative of \( f(x)=5-8 x-x^{2} \) on your calculator. At what value of \( x \) do the tangent lines switch from having positive slopes to negative slopes? \( x=[?] \)

To graph the derivative of the function \( f(x) = 5 - 8x - x^2 \), first, we need to find the derivative, which is \( f'(x) = -8 - 2x \). When you plot this derivative, you'll see a straight line with a negative slope. To find where the tangent lines switch from positive to negative slopes, set the derivative equal to zero: \( -8 - 2x = 0 \). Solving for \( x \), we get \( x = -4 \). This means that at \( x = -4 \), the slope changes from positive to negative, indicating a maximum point on the original function.