Find the inlervals on which the graph of I is concave upward, the intorvals on which tho graph off is concave downward, and the inflection points \[ f(x)=x^{3}-18 x^{2}+2 x+2 \] For what inlerval(s) of \( x \) is the graph of \( I \) concave upward? Select the correct duice bebw and, if necessary, fill in the answer box lo complete your chnice A. \( (6, \infty) \) (Type your answer in inleval notalion. Type an exacl answer. Uso a comna to separato answers as noodod.) B. The graph is never concave upward. Fot what inlerval(\$) of \( x \) is the graph of 1 concave downward? Select the conect choica below and, il necessary, fill in the answer hox to complete your choico. A. \( (-\infty, 6) \) (Type your answer in inlerval notation. Type an viacl answer. Use a comma to soparate answers as needed.) B. The graph is never concave downward. Delermine the \( x \), \( y \) coordinates of the inflection point of the graph of \( f \). Solect the correct choice below and, if necessary, fil in the answer box to complete your choice A. \( \square \) (Typs an ondered pair.) B. There are no inlloction ponts.
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To determine where the graph of the function \( f(x) = x^3 - 18x^2 + 2x + 2 \) is concave upward or downward, we need to analyze its second derivative. The first derivative \( f'(x) = 3x^2 - 36x + 2 \), and the second derivative \( f''(x) = 6x - 36 \). Setting the second derivative to zero gives us \( x = 6 \). Thus, the graph is concave upward on \( (6, \infty) \) and concave downward on \( (-\infty, 6) \). The inflection point occurs where the concavity changes, which is at the coordinate \( (6, f(6)) \). We calculate \( f(6) = 6^3 - 18 \cdot 6^2 + 2 \cdot 6 + 2 = -86 \). Therefore, the inflection point is \( (6, -86) \). Concavity is all about the change! Fun fact: concavity relates to how the slopes of tangent lines behave on the graph—if they are getting steeper (upward) or less steep (downward). This concept is widely applicable in fields like economics, where understanding whether returns on investment accelerate or decelerate can significantly impact strategy and planning.
