Final answer:
To find the absolute maximum, substitute x with y² in the function, take the derivative, and evaluate at critical points and endpoints, then compare the values.
Step-by-step explanation:
To find the absolute maximum of the function f(x, y) = x² - y² on the curve x = y², within the specified range 0 ≤ y ≤ 2, we will first substitute x with y² to get the function in terms of one variable, resulting in f(y) = y´ - y². We then find the derivative of this function to get f'(y) = 4y³ - 2y. Setting this derivative equal to zero will give us the critical points, which are y = 0 and y = √0.5. We evaluate the function at these points and the endpoints of the interval to find the maximum value. In this case, plugging the values y = 0, y = √0.5, and y = 2 into the function f(y) will give us the potential maximum values. Comparing these values will provide us with the absolute maximum value within the given range.