Final answer:
To find the absolute max and min values of f(x) = ln(x² - 3x + 6) on [-2, 1], calculate the derivative, find critical points, and evaluate f(x) at those points along with the interval's endpoints.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x) = ln(x² - 3x + 6) on the interval [-2, 1], we need to check the values of the function at the critical points and the endpoints of the interval. First, we'll find the derivative of the function, f'(x), and set it equal to zero to find the critical points within the interval. After that, we must evaluate f(x) at the critical points and at the endpoints, x = -2 and x = 1.
First, let's find the derivative: f'(x) = ⅔ (2x - 3) / (x² - 3x + 6). Setting the numerator equal to zero gives us a critical point at x = ⅔.
We then evaluate the function at x = ⅔, f(-2), and f(1). Since the natural logarithm is undefined for non-positive arguments, and the quadratic expression x² - 3x + 6 is always positive for all real x, there are no issues of domain restrictions within the interval.
After evaluating f(x) at these points, the largest value will be the absolute maximum and the smallest value will be the absolute minimum of the function on the given interval [-2, 1].