Although Fig. 11.10 specifically illustrates the one-variable case, the definitions of S and S in (11.27) and (1 1.28) are not limited to functions of a single variable. They are equally valid if we interpret x to be a vector, i.e., let x = (X1, . . . .x,). In that case, however, (1 1.27) and (1 1.28) will define convex sets in the n-space instead, It is important to re- member that while a convex function implies (1 1.27), and a concave function implies (11.28), the converse is not true -for (1 1.27) can also be satisfied by a nonconvex function and (1 1.28) by a nonconcave function. This is discussed further in Sec. 12.4. ERCISE 11.5 1. Use (11.20) to check whether the following functions are concave, convex, strictly con- cave, strictly convex, or neither: (0) z= x2 (b) z= x7 + 2x} (c) z=2x2 - xy+ yz 2. Use (11.24) or (11.24') to check whether the following functions are concave, convex, strictly concave, strictly convex, or neither: (a) z=-x2 (b) z = (x1 - x2) 2 (C) 2 = -xy 3. In view of your answer to Prob. 2c, could you have made use of Theorem Ill of this section to compartmentalize the task of checking the function z = 2x2 - xy + y in Prob. 1c? Explain your answer. acer C $ % A 5 6