1. Look at Morse's lemma (contained in Milnor's book at a page p < 25).
2. Show that x_1^2 + ... + x_n^2 is minimal at (x_1, ..., x_n) = (0, ..., 0).
Hint: the square of a non-zero real number is stricly positive.
3. Show that - (x_1^2 + ... + x_n^2) is maximal at (x_1, ..., x_n) = (0...