Find the intervals on which \( f(x) \) is increasing, the intervals on which \( f(x) \) is decreasing, and the local extrema. \( f(x)=-3 x^{2}-30 x-21 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is increasing on ( \( -\infty,-5 \) ). (Type your answer in interval notation. Type integers or simplified fractions. Use a comma to separate answers as needed.) The function is never increasing. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is decreasing on (Type your answer in interval notation. Type integers or simplified fractions. Use a comma to separate answers as needed.) B. The function is never decreasing.

**Step 1. Find the derivative of \( f(x) \).** We have \[ f(x)=-3x^2-30x-21. \] Differentiate with respect to \( x \): \[ f'(x) = \frac{d}{dx}\left(-3x^2-30x-21\right) = -6x-30. \] **Step 2. Find the critical point by setting \( f'(x) = 0 \).** Set \[ -6x-30=0. \] Solve for \( x \): \[ -6x = 30 \quad \Longrightarrow \quad x=-5. \] **Step 3. Determine the intervals on which \( f(x) \) is increasing or decreasing.** Write \( f'(x) \) in factored form: \[ f'(x)=-6(x+5). \] - For \( x < -5 \): The term \( x+5 < 0 \), so \[ -6(x+5) > 0. \] Hence, \( f'(x) > 0 \), and \( f(x) \) is increasing. - For \( x > -5 \): The term \( x+5 > 0 \), so \[ -6(x+5) < 0. \] Hence, \( f'(x) < 0 \), and \( f(x) \) is decreasing. Thus, the function is increasing on \((-\infty,-5)\) and decreasing on \((-5,\infty)\). **Step 4. Identify the local extrema.** Since \( f(x) \) changes from increasing to decreasing at \( x=-5 \), there is a local maximum at \( x = -5 \). Find the local maximum value: \[ f(-5) = -3(-5)^2 - 30(-5) - 21. \] Calculate: \[ f(-5) = -3(25) + 150 - 21 = -75 + 150 - 21 = 54. \] So, there is a local maximum at \((-5, 54)\). **Final Answers:** - The function is increasing on \((-\infty, -5)\). - The function is decreasing on \((-5, \infty)\). - There is a local maximum at \((-5, 54)\).