Final answer:
To find the relative and absolute extrema of the function g(x) = x^4 - 4x on the domain [-3, 1], we can follow these steps: take the derivative, find critical points, evaluate function at endpoints, compare values to determine extrema. The extrema for this function on the given domain are as follows: absolute maximum at (81, 6561), relative minimum at (-3, 81), and another relative minimum at (1, -3).
Step-by-step explanation:
To find the relative and absolute extrema of the function g(x) = x4 - 4x on the domain [-3, 1], we can follow these steps:
- Take the derivative of g(x) to find critical points.
- Find the endpoints of the domain and evaluate g(x) at those points.
- Compare the values of g(x) at critical points and endpoints to determine the relative and absolute extrema.
Let's proceed with these steps:
- Taking the derivative of g(x), we get g'(x) = 4x3 - 4.
- Setting g'(x) = 0, we solve 4x3 - 4 = 0 to find the critical points. Simplifying, we get x3 - 1 = 0, which factors as (x-1)(x2 + x + 1) = 0. Consequently, the critical point is x = 1.
- The domain [-3, 1] has two endpoints, -3 and 1. Evaluating g(x) at these points, we find g(-3) = 81 and g(1) = -3.
- Next, we compare the values of g(x) at the critical point and endpoints. We find that g(1) = -3 is the smallest value, g(81) = 6561 is the largest value, and g(-3) = 81 is another relative minimum.
Therefore, the relative and absolute extrema of the function g(x) = x4 - 4x on the domain [-3, 1] are:
- Absolute maximum: g(81) = 6561 at x = 81
- Relative minimum: g(-3) = 81 at x = -3
- Relative minimum: g(1) = -3 at x = 1