If g is a differentiable function such that g(x) is less than 0 for all real numbers x and if f'(x)=(x2-4)g(x), which of the following is true? f has a relative maximum at x=-2 and a relative minimum at x=2, f has a relative minimum at x=-2 and has a relative maximum at x=2, f has relative minima at x=-2 and at x=2, f has a relative maxima at x=-2 and at x=2, or it cannot be determined if f has any relative extrema
To determine the relative extrema of the function f(x), we need to examine the sign of its derivative, f'(x). Given f'(x) = (x^2 - 4)g(x) and the fact that g(x) < 0 for all real numbers x, we can analyze the behavior of f'(x) and make conclusions about the extrema of f(x).
1. For x < -2: Since g(x) < 0, and (x^2 - 4) is positive, f'(x) will be positive. This indicates that the function f(x) is increasing on the interval x < -2.
2. For -2 < x < 2: Since g(x) < 0, and (x^2 - 4) is negative, f'(x) will be negative. This implies that the function f(x) is decreasing on the interval -2 < x < 2.
3. For x > 2: Since g(x) < 0, and (x^2 - 4) is positive, f'(x) will be positive. This indicates that the function f(x) is increasing on the interval x > 2.
Based on this analysis, we can conclude that f(x) has a relative minimum at x = -2 and a relative maximum at x = 2. Therefore, the correct statement is: f has a relative minimum at x = -2 and a relative maximum at x = 2.