Find the intervals on which \( f(x) \) is increasing and the intervals on which \( f(x) \) is decreasing. Then sketch the graph. Add horizontal tangent lines. \( f(x)=2-27 x+9 x^{2}-x^{3} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The function is increasing on (Type your answer using interval notation. Use a comma to separate answers as needed.) B. The function is never increasing.

To determine the intervals where the function \( f(x) = 2 - 27x + 9x^2 - x^3 \) is increasing or decreasing, we first need to compute the derivative \( f'(x) \). Calculating the derivative: \[ f'(x) = -27 + 18x - 3x^2 \] Next, we set the derivative equal to zero to find the critical points: \[ 0 = -3x^2 + 18x - 27 \] Dividing through by -3 gives: \[ 0 = x^2 - 6x + 9 \] Factoring the quadratic results in: \[ 0 = (x - 3)^2 \] So, \( x = 3 \) is a double root and thus a critical point. To find the intervals of increasing and decreasing behavior, we can use a sign chart to test the intervals around the critical point \( x = 3 \). - For \( x < 3 \) (choose \( x = 0 \)): \[ f'(0) = -27 + 0 + 0 = -27 < 0 \] So \( f(x) \) is decreasing on \( (-\infty, 3) \). - For \( x > 3 \) (choose \( x = 4 \)): \[ f'(4) = -27 + 72 - 48 = -3 < 0 \] So \( f(x) \) is also decreasing on \( (3, \infty) \). Since \( f'(x) < 0 \) for both sides of the critical point, the function is always decreasing. The function is decreasing on \( (-\infty, 3) \) and \( (3, \infty) \), with no intervals of increasing behavior. A. The function is increasing on: (empty set) B. The function is never increasing. To sketch the graph of the function, we'd find the y-intercept by evaluating \( f(0) \): \[ f(0) = 2 \] To find other critical behavior, we can evaluate \( f(3) \): \[ f(3) = 2 - 27(3) + 9(3^2) - (3^3) = 2 - 81 + 81 - 27 = -25 \] Thus, we see the function pivots near the point \( (3, -25) \). The end behavior also paints a clear picture given it's a cubic polynomial: as \( x \to -\infty \), \( f(x) \to \infty \) and as \( x \to \infty \), \( f(x) \to -\infty \). Now that we’re set, add horizontal tangent lines at \( x = 3 \) where there’s a double root and no increase. Isn’t math a bit like a plot twist? Just when you think the function might throw a party with an increase, it quietly declines instead! Happy sketching!