For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither, b) identify intervals where the function is increasing or decreasing, c) give the coordinates of any points of inflection, d) identify intervals where the function is concave up or concave down, and e) sketch the graph. \( g(x)=x^{3}-9 x^{2}+15 x+5 \)

To analyze the function \( g(x) = x^3 - 9x^2 + 15x + 5 \), we will follow the steps outlined in your request. ### a) Critical Points and Classification 1. **Find the first derivative**: \[ g'(x) = 3x^2 - 18x + 15 \] 2. **Set the first derivative to zero to find critical points**: \[ 3x^2 - 18x + 15 = 0 \] Dividing the entire equation by 3: \[ x^2 - 6x + 5 = 0 \] Factoring: \[ (x - 1)(x - 5) = 0 \] Thus, the critical points are: \[ x = 1 \quad \text{and} \quad x = 5 \] 3. **Classify the critical points using the second derivative**: \[ g''(x) = 6x - 18 \] Evaluate the second derivative at the critical points: - For \( x = 1 \): \[ g''(1) = 6(1) - 18 = 6 - 18 = -12 \quad (\text{concave down} \Rightarrow \text{relative maximum}) \] - For \( x = 5 \): \[ g''(5) = 6(5) - 18 = 30 - 18 = 12 \quad (\text{concave up} \Rightarrow \text{relative minimum}) \] **Summary of critical points**: - \( (1, g(1)) = (1, 1^3 - 9(1^2) + 15(1) + 5) = (1, 1 - 9 + 15 + 5) = (1, 12) \) → Relative Maximum - \( (5, g(5)) = (5, 5^3 - 9(5^2) + 15(5) + 5) = (5, 125 - 225 + 75 + 5) = (5, -20) \) → Relative Minimum ### b) Intervals of Increase and Decrease 1. **Find the sign of \( g'(x) \)**: - The critical points divide the number line into intervals: \( (-\infty, 1) \), \( (1, 5) \), and \( (5, \infty) \). - Test points in each interval: - For \( x = 0 \) in \( (-\infty, 1) \): \[ g'(0) = 3(0)^2 - 18(0) + 15 = 15 > 0 \quad (\text{increasing}) \] - For \( x = 3 \) in \( (1, 5) \): \[ g'(3) = 3(3)^2 - 18(3) + 15 = 27 - 54 + 15 = -12 < 0 \quad (\text{decreasing}) \] - For \( x = 6 \) in \( (5, \infty) \): \[ g'(6) = 3(6)^2 - 18(6) + 15 = 54 - 108 + 15 = -39 < 0 \quad (\text{decreasing}) \] **Summary of intervals**: - Increasing on \( (-\infty, 1) \) - Decreasing on \( (1, 5) \) and \( (5, \infty) \) ### c) Points of Inflection 1. **Find the second derivative**: \[ g''(x) = 6x - 18 \] 2. **Set the second derivative to zero**: \[ 6x - 18 = 0 \quad \Rightarrow \quad x = 3 \] 3. **Evaluate \( g(3) \)**: \[ g(3) = 3^3 - 9(3^2) + 15(3) + 5 = 27 - 81 + 45 + 5 = -4 \] Thus, the point of inflection is \( (3, -4) \). ### d) Intervals of Concavity 1. **Determine the sign of \( g