What is the point of inflection of the function \( y=-4(x+7)^{3}-11 \) ? (1 point) \( (-7,-11) \) \( (-7,11) \) \( (7,-11) \) \( (7,11) \)

To find the point of inflection of the function \( y = -4(x+7)^{3}-11 \), we need to determine where the second derivative changes sign. Taking the first derivative gives us \( y' = -12(x+7)^{2} \). The second derivative is \( y'' = -24(x+7) \). Setting the second derivative equal to zero, we find that \( y'' = 0 \) at \( x = -7 \). Substituting \( x = -7 \) back into the original function yields \( y = -4(-7+7)^{3} - 11 = -11 \). Therefore, the point of inflection is \( (-7, -11) \). Is this choice in the options? Yes! The answer is \( (-7, -11) \). Taking the derivative of a function is like peeking behind the curtain—once you understand the changes in slope, you get to uncover the hidden secrets of its behavior! It’s where the graph goes from curving up to curving down (or vice versa) that you find those magical points of inflection! Additionally, when identifying points of inflection, remember that not all points where the second derivative equals zero will be inflection points; you always need to check if the concavity actually changes on either side of the point!