Final answer:
Critical points occur where the first derivative of a function is zero or undefined, excluding points where the original function is undefined. To find these for f(x) = -2/((x+5)(x+1)), calculate and analyze the function's first and second derivatives, also considering the signs of the derivatives around the critical points to determine stability and relative maxima or minima.
Step-by-step explanation:
To find the critical points of the function f(x) = -2/((x+5)(x+1)), we need to first determine where the function's derivative is zero or undefined. The derivative of the function, f'(x), can help us identify these points.
To locate the critical points, calculate the derivative of f(x), set f'(x) equal to zero, and solve for x. Also consider where the derivative is undefined, which occurs where the denominator is zero. In this case, since the denominator is the same as that of f(x), the derivative will be undefined at x = -5 and x = -1. However, since these are the points where the original function f(x) is undefined, they cannot be considered critical points according to the usual definition.
A critical point that is a relative maximum or minimum or represents stable or unstable equilibrium can be identified by examining the signs of the first and second derivatives of the function around the critical points. If the first derivative changes signs at a critical point, and the second derivative is positive, this indicates a relative minimum and a stable equilibrium; if it changes signs and the second derivative is negative, it suggests a relative maximum and an unstable equilibrium.