Q.9; Describe the increasing and decreasing intervals in the branches of the function \[ g(x)=-1 /\left(x^{2}+3\right) \]

Step 1 ：Given the function \(g(x) = -1/(x^{2} + 3)\)

Step 2 ：Find the derivative of the function to get \(g'(x) = 2x/(x^{2} + 3)^{2}\)

Step 3 ：Determine the critical points by setting the derivative equal to zero or undefined. The critical point is \(x = 0\)

Step 4 ：Test the intervals on the number line to determine where the function is increasing or decreasing. The derivative of the function is always positive for \(x < 0\) and always negative for \(x > 0\)

Step 5 ：Conclude that the function is increasing on the interval \((-\infty, 0)\) and decreasing on the interval \((0, \infty)\)

Step 6 ：Final Answer: The increasing and decreasing intervals are \(\boxed{(-\infty, 0)}\) and \(\boxed{(0, \infty)}\) respectively