Find the intervals on which the graph of \( f \) is concave upward, the intervals on which the graph of \( f \) is concave downward, and the inflection points. \( f(x)=x^{3}-15 x^{2}+2 x+3 \) For what interval(s) of \( x \) is the graph of \( f \) concave upward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The graph is never concave upward. For what interval(s) of \( x \) is the graph of \( f \) concave downward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. B. The graph is never concave downward. Determine the \( x, y \) coordinates of the inflection point of the graph of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Type an ordered pair.) (Typerval notation. Type an exact answer. Use a comma to separate answers as needed.) (T)

To determine the intervals of concavity and the inflection points of the function \( f(x) = x^{3} - 15x^{2} + 2x + 3 \), we first need to find the second derivative of \( f \). 1. **First Derivative**: \[ f'(x) = 3x^{2} - 30x + 2 \] 2. **Second Derivative**: \[ f''(x) = 6x - 30 \] 3. **Setting the second derivative to zero** to find inflection points: \[ 6x - 30 = 0 \\ x = 5 \] 4. **Testing intervals for concavity**: - For \( x < 5 \), pick \( x = 0 \): \[ f''(0) = 6(0) - 30 = -30 \quad \text{(concave downward)} \] - For \( x > 5 \), pick \( x = 6 \): \[ f''(6) = 6(6) - 30 = 6 \quad \text{(concave upward)} \] The graph of \( f \) is: - **Concave upward** on the interval: \( (5, \infty) \) - **Concave downward** on the interval: \( (-\infty, 5) \) Now, we find the coordinates of the inflection point by plugging \( x = 5 \) back into \( f(x) \): \[ f(5) = 5^3 - 15(5^2) + 2(5) + 3 = 125 - 375 + 10 + 3 = -237 \] Thus, the inflection point is: \( (5, -237) \). **Final Answers**: For concave upward: \( (5, \infty) \) For concave downward: \( (-\infty, 5) \) Inflection point: \( (5, -237) \)