Final answer:
The question involves finding the extremes of a given function on the constraint of a circle, which is a problem of optimization in mathematics.
Step-by-step explanation:
The question asks to find the maximum and minimum values of the function F (x, y) = x² - xy - 1/2y subjected to the constraint x² + y² =1. This is a problem of optimization with a constraint, which can typically be solved using the method of Lagrange multipliers.
However, because the constraint is the equation of a circle, this particular problem can also be solved by substituting for y using the constraint and finding the extremal values directly or by parameterizing the circle using trigonometric identities and then finding the extrema.