Find the intervals on which \( f(x) \) is increasing, the intervals on which \( f(x) \) is decreasing, and the local extrema. \( f(x)=-3 x^{2}-30 x-21 \) 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.) B. The function is never increasing. A. The function is decreasing on \( (-5, \infty) \). (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. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Type integers or simplified fractions.) A. The function has a local minimum \( f(\square)=\square \), and no local maximum B. The function has a local maximum \( f(\square)=\square \) and a local minimum \( f(\square)=\square \), fill in the answer box to complete your choice. C. The function has a local maximum \( f(\square)=\square \), and no local minimum. D. The function has no local extrema.

\( f(x) = -3x^2 - 30x - 21 \) is a quadratic function with a negative coefficient for \( x^2 \), so it opens downward. This means it will have a local maximum at the vertex and is decreasing on one side of the vertex. 1. To determine where the function is increasing or decreasing, we compute the first derivative: \[ f'(x) = \frac{d}{dx}(-3x^2 - 30x - 21) = -6x - 30. \] 2. Set the derivative equal to zero to find the critical point: \[ -6x - 30 = 0 \quad \Longrightarrow \quad x = -5. \] 3. Analyze the sign of \( f'(x) \): - For \( x < -5 \): Choose \( x = -6 \). \[ f'(-6) = -6(-6) - 30 = 36 - 30 = 6 \quad (\text{positive}). \] Thus, \( f(x) \) is increasing on \( (-\infty, -5) \). - For \( x > -5 \): Choose \( x = -4 \). \[ f'(-4) = -6(-4) - 30 = 24 - 30 = -6 \quad (\text{negative}). \] Thus, \( f(x) \) is decreasing on \( (-5, \infty) \). 4. Since the function changes from increasing to decreasing at \( x = -5 \), there is a local maximum at \( x = -5 \). Evaluate \( f(-5) \) to find the maximum value: \[ f(-5) = -3(-5)^2 - 30(-5) - 21. \] \[ (-5)^2 = 25 \quad \Rightarrow \quad f(-5) = -3(25) + 150 - 21 = -75 + 150 - 21 = 54. \] 5. Summary of the answers: - The function is increasing on \( (-\infty, -5) \). - The function is decreasing on \( (-5, \infty) \). - The function has a local maximum \( f(-5)=54 \) and no local minimum. Thus, the correct selections are: A. The function is increasing on \( (-\infty,-5) \). A. The function is decreasing on \( (-5,\infty) \). C. The function has a local maximum \( f(-5)=54 \), and no local minimum.