Find the nlarvals on which the graph of \( f \) is concavo upward, the inlervals on which tie graph of 1 is cencrave downard, and the inflection points. \[ 1(x)=x^{3}-18 x^{2}+2 x+2 \] For what inlerval(s) of x is the graph of I concave upward? Select the correct choice below and, if necessay y , fill in the answet box lo complele your chaice A. \( (6, \infty) \) (Typo your answer in inlerval notalion. Typo an couct answer. Use a curnma to separale answers as noeded.) B. The graph is never concave upward. For what interve (s) of x is the grapt of 1 concave downward? Select the coned chnice below and, if necossary fill in the answur bux to compete your choce. A. \( (-\infty, 6) \) (Typo your answer in interval notalion. Typo an oxaci answer. Use a comma lo separatu answors as neoded.) B. The graph is never concave downward. Determine the \( \mathrm{x}, \mathrm{y} \) cuordinatos of the inflection point ol the graph of L. Select the corect choice bolow and if nencessery, fill in the answar box to complate your choic A. \( \square \) (Type an ordered pair) B. There are no inflection points

1. Compute the first and second derivatives of \( f(x)=x^3-18x^2+2x+2 \). - The first derivative is \[ f'(x)=3x^2-36x+2. \] - The second derivative is \[ f''(x)=6x-36. \] 2. Determine the concavity by analyzing the sign of \( f''(x) \). - For concave upward, we need \( f''(x)>0 \): \[ 6x-36>0 \quad\Longrightarrow\quad 6x>36 \quad\Longrightarrow\quad x>6. \] Thus, the function is concave upward on \( (6,\infty) \). - For concave downward, we need \( f''(x)<0 \): \[ 6x-36<0 \quad\Longrightarrow\quad 6x<36 \quad\Longrightarrow\quad x<6. \] Thus, the function is concave downward on \( (-\infty,6) \). 3. Find the inflection point by solving \( f''(x)=0 \): \[ 6x-36=0 \quad\Longrightarrow\quad x=6. \] Substitute \( x=6 \) into \( f(x) \) to find the \( y \)-coordinate: \[ f(6)=6^3-18(6)^2+2(6)+2. \] Calculate step by step: \[ 6^3=216,\quad (6)^2=36,\quad 18(36)=648,\quad 2(6)=12. \] Thus, \[ f(6)=216-648+12+2= (216-648) + 14= -432+14= -418. \] So, the inflection point is \( (6,-418) \). 4. Answer the given questions: - For what interval(s) of \( x \) is the graph of \( f \) concave upward? Correct choice: **A. \( (6, \infty) \)** - For what interval(s) of \( x \) is the graph of \( f \) concave downward? Correct choice: **A. \( (-\infty, 6) \)** - Determine the \( x, y \) coordinates of the inflection point of the graph of \( f \). Correct choice: **A. \( (6,-418) \)**