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 (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.

To determine the intervals where \( f(x) \) is increasing or decreasing, we start by finding the derivative \( f'(x) \): \[ f'(x) = -6x - 30 \] To find critical points, we set the derivative to zero: \[ -6x - 30 = 0 \] \[ -6x = 30 \] \[ x = -5 \] Now, we use this critical point to test the intervals. The derivative \( f'(x) \) is a linear function and will change signs around \( x = -5 \): - For \( x < -5 \), pick \( x = -6 \): \[ f'(-6) = -6(-6) - 30 = 36 - 30 = 6 \quad (\text{positive, so } f(x) \text{ is increasing}) \] - For \( x > -5 \), pick \( x = -4 \): \[ f'(-4) = -6(-4) - 30 = 24 - 30 = -6 \quad (\text{negative, so } f(x) \text{ is decreasing}) \] Now we can summarize: A. The function is increasing on \( (-\infty, -5) \). B. The function is decreasing on \( (-5, \infty) \). To find the local extremum, we evaluate \( f(-5) \): \[ f(-5) = -3(-5)^2 - 30(-5) - 21 = -3(25) + 150 - 21 = -75 + 150 - 21 = 54 \] So, there is a local maximum at \( (-5, 54) \). In summary, A. The function is increasing on \( (-\infty, -5) \). B. The function is never increasing is incorrect.