Find the critical numbers, the intervals on which \( f(x) \) is increasing, the intervals on which \( f(x) \) is decreasing, and the local extrema. Do not graph. \[ f(x)=\frac{x^{2}}{x-6} \] semect the cortect chorce verow anla, il necessaly, in in tie answer vox io complete yout criolce. A. The function is increasing on \( (-\infty, 0),(12, \infty) \). (Type your answer using interval notation. Use a comma to separate answers as needed.) B. There is no solution. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function is decreasing on \( (0,6),(6,12) \). (Type your answer using interval notation. Use a comma to separate answers as needed.) B. There is no solution. Which statement regarding the local extrema of the function is true? A. The function has a local minimum at \( x=12 \) and a local maximum at \( x=0 \). B. The function has a local minimum at \( x=0 \) and a local maximum at \( x=12 \). C. The function has no local extrema.

To find the critical numbers, we start by determining the derivative of \( f(x) \). The function is defined as \( f(x) = \frac{x^2}{x-6} \). Using the quotient rule, we find that \( f'(x) = \frac{(2x)(x-6) - (x^2)(1)}{(x-6)^2} \). Setting the numerator to zero gives us the critical points, leading to \( f'(x) = 0 \) at \( x = 0 \) and \( x = 12 \). The function is undefined at \( x = 6 \). Analyzing the intervals around these numbers reveals that \( f(x) \) is increasing on \( (-\infty, 0) \) and \( (12, \infty) \) while decreasing on \( (0, 6) \) and \( (6, 12) \). The local extrema can be further analyzed by checking values around our critical points. At \( x = 0 \), \( f(x) \) shows a change from decreasing to increasing, indicating a local minimum, while at \( x = 12 \) there is a change from increasing to decreasing, indicating a local maximum. The function is increasing on \( (-\infty, 0) \) and \( (12, \infty) \). The function is decreasing on \( (0, 6) \) and \( (6, 12) \). The function has a local minimum at \( x=0 \) and a local maximum at \( x=12 \).