Final answer:
To find the absolute maximum and minimum of the function g(x) on the interval [0,20], we find the critical points and evaluate the function at these points and at the interval's endpoints. The absolute minimum is 4 at x = 1, and the absolute maximum is 7946 at x = 20.
Step-by-step explanation:
The question asks us to find the absolute maximum and absolute minimum values of a given function, g(x)=x³−3x+6, on a specific interval. To do this, we must evaluate the function at its critical points within the interval and also at the endpoints of the interval. Critical points occur where the first derivative of the function is zero or undefined. Since the function is a polynomial, it's differentiable everywhere, so we only need to set the derivative equal to zero to find the critical points.
First, we find the derivative of g(x):
g'(x) = 3x² - 3
To find critical points, solve for x when g'(x) = 0:
0 = 3x² - 3
x² = 1
x = ± 1
These x-values are within our interval [0,20]. We then evaluate the function at x = 1, x = -1, x = 0, and x = 20:
- g(1) = 1³ - 3·1 + 6 = 4
- g(-1) = (-1)³ - 3(-1) + 6 = 8
- g(0) = 0³ - 3·0 + 6 = 6
- g(20) = 20³ - 3·20 + 6 = 8000 - 60 + 6 = 7946
The absolute minimum value is 4 at x = 1, and the absolute maximum value is 7946 at x = 20.