(c) global maximum of at , global minimum of at .
(f) local maximum at , local minimum at .
(j) global minimum at .
(n) no extrema
Proof. If , then is a critical point of at which the derivative is zero, so we may apply the second derivative test to . If , then has an extremum at , so that the derivative of changes at and hence the concavity changes at . Thus is an inflection point of .
(b) P1 should replace in with the value that P2 is going to use, and minimize the resulting , getting .
(c) P1 should minimize with respect to , getting .
(d) P2 should replace in with the value that P1 is going to use and maximize the resulting , getting .
(e) Each player has to ``go first" so P1 chooses and P2 chooses . (The resulting score is 0, so the game is uninteresting, even though it is fair.)