Fine the invervals on which the graph of I is cancavo upwate, the invervats on which the graph of I is concave dowerward, and the innocion points \[ f(x)=x^{3}-6 x^{2}+13 x+4 \] For what interval(s) of \( x \) is the graph off concave upward? Setect ine conisct choice below and, if nocessary, fill in the answer box to complete your dwice. A. \( \square \) (Type your ancwor in intorval nolabon. Type an oxact arswel. Use a cormura to soperelo answors as needind) B. The graph is novar concavo upward. For what interval( \( s \) ) of \( x \) is the graph of I concave dowmwa'd Selat the corract choice boljw and, if necessary, fill in the answor bor to complete your choice. A. \( \square \) (Trpo your answer in inlerval nolation. Type an exact answer. Uso a conma ld separale answers as neodud) B. The graph is never concave downward. Determinn thox, y coordnates of the inflection point of the graph of I. Select tie corred choice below and, inecossary, fill in the enswer box to ccmpleta your cheice.

1. Compute the first and second derivatives of \[ f(x) = x^3 - 6x^2 + 13x + 4. \] The first derivative is \[ f'(x) = 3x^2 - 12x + 13. \] The second derivative is \[ f''(x) = 6x - 12. \] 2. Determine the inflection point by setting the second derivative equal to zero: \[ 6x - 12 = 0 \quad \Longrightarrow \quad x = 2. \] Substitute \( x = 2 \) into \( f(x) \) to find the \( y \)-coordinate: \[ f(2) = 2^3 - 6(2)^2 + 13(2) + 4 = 8 - 24 + 26 + 4 = 14. \] Thus, the inflection point is at \[ (2, 14). \] 3. Determine the intervals of concavity: - For \( x < 2 \): Since \[ f''(x) = 6x - 12 \quad \text{and if} \quad x < 2 \quad \text{then} \quad 6x - 12 < 0, \] the graph is concave downward on the interval \[ (-\infty, 2). \] - For \( x > 2 \): Since \[ f''(x) = 6x - 12 \quad \text{and if} \quad x > 2 \quad \text{then} \quad 6x - 12 > 0, \] the graph is concave upward on the interval \[ (2, \infty). \] Final Answers: - For what interval(s) of \( x \) is the graph concave upward? Choice A: \( (2, \infty) \). - For what interval(s) of \( x \) is the graph concave downward? Choice A: \( (-\infty, 2) \). - The inflection point of the graph is at \( (2, 14) \).