Final answer:
The extrema of the function g(x) = 6x³ − 72x with domain [−4, 4] include an absolute maximum of 384 at x = 4 and an absolute minimum of -384 at x = -4, with relative extrema at x = -2 (minimum) and x = 2 (maximum).
Step-by-step explanation:
The student has asked to find the extrema of the function g(x) = 6x³ − 72x with the domain [−4, 4]. To find the relative extrema, we will need to calculate the derivative of g(x) and set it equal to zero to solve for x.
The derivative of g(x) is g'(x) = 18x² - 72. Setting g'(x) = 0 gives us x² = 4, or x = ±2. Evaluating g(x) at x = ±2 gives us the relative extrema. For the absolute extrema, we will also evaluate g(x) at the endpoints of the domain, x = -4 and x = 4.
To find out whether these critical points are maxima or minima, we can use the First Derivative Test or the Second Derivative Test. The second derivative is g''(x) = 36x. Since g''(±2) is positive, x = -2 is a relative minimum, and since it's negative at x = 2, that's a relative maximum.
The values of g(x) at the critical points and endpoints are as follows:
- g(-4) = -384
- g(-2) = 48
- g(2) = -48
- g(4) = 384
Hence, the function has an absolute maximum of 384 at x = 4 and an absolute minimum of -384 at x = -4; there are relative extrema at x = -2 (minimum) and x = 2 (maximum).