Find tion \( x \) and \( y \) coordinates of all infoction ponts \[ f(x)=x^{3}+3 x x^{2} \] What is/are the inlledinn poinl(s)? Seloct the corruci choice below and, I necessary, fill in the answor box lo complete your choce A. The inflection point(s) is/are \( \square \) (Typo an ordured par. Uso a comma lo soparalo answors as needed ) B. Thore are no inflection points. mple Get more help -

To find the inflection points of the function \( f(x) = x^3 + 3x^2 \), we first need to compute the second derivative of the function. 1. **First Derivative**: \[ f'(x) = 3x^2 + 6x \] 2. **Second Derivative**: \[ f''(x) = 6x + 6 \] 3. **Set the Second Derivative to Zero**: \[ 6x + 6 = 0 \Rightarrow x = -1 \] To find the corresponding \( y \)-coordinate, we substitute \( x = -1 \) back into the original function: \[ f(-1) = (-1)^3 + 3(-1)^2 = -1 + 3 = 2 \] Thus, the inflection point is at \( (-1, 2) \). A. The inflection point(s) is/are \( (-1, 2) \). --- Dive into the fascinating world of calculus further! Inflection points are critical because they indicate a change in the curvature of the graph, helping you understand the shape and behavior of functions in real-world scenarios like engineering and economics. --- If you're looking for practical problem-solving tips, remember to double-check your derivatives and sign changes when determining inflection points. Common mistakes include missing a factor of zero or overlooking changes in concavity, which could lead to incorrect conclusions about the function's behavior.