Final answer:
To find the absolute maximum and minimum values of f(x) = x/(x² - x + 1) on [0,3], we determine critical points by setting the derivative to zero, evaluate f(x) at these points and the endpoints, then compare these values.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x) = x/(x² - x + 1) on the interval [0,3], we need to evaluate the function at critical points within the interval and at the endpoints. The critical points are where the derivative of the function equals zero or does not exist. To find the derivative, use the quotient rule:
f'(x) = [(x² - x + 1)(1) - x(2x - 1)] / (x² - x + 1)²
Setting the numerator equal to zero and solving for x will give the critical points. Next, evaluate f(x) at the critical points and at the endpoints of the interval, x=0 and x=3. The largest value will be the absolute maximum and the smallest value will be the absolute minimum on the given interval.
Since the function is a ratio of polynomials, it is continuous and differentiable everywhere, so there are no points where the derivative does not exist between 0 and 3. However, we must also check the endpoints to determine the absolute extremum values.
The graph of f(x) can be labeled with f(x) and x, and the axes should be scaled with the maximum and minimum values of f(x) found in this interval.