5. Assume that the set of alternatives is X=R+2 and consider a consumer with utility function u(x1,x2)=x1+lnx2. Fix prices (p1,p2)>0 and wealth w>0. (a) Are the consumer's preferences locally nonsatiated? (b) Are the consumer's preferences convex? (c) Write down the consumer's utility-maximisation problem. (d) Write down the Lagrangean and derive the first-order necessary conditions. (e) Does there exist a solution x to the first-order necessary conditions with x1=0 ? (f) Does there exist a solution x to the first-order necessary conditions with x2=0 ? (g) Derive the consumer's demand correspondence x(p,w). 5. Assume that the set of alternatives is X=R+2 and consider a consumer with utility function u(x1,x2)=x1+lnx2. Fix prices (p1,p2)>0 and wealth w>0. (a) Are the consumer's preferences locally nonsatiated? (b) Are the consumer's preferences convex? (c) Write down the consumer's utility-maximisation problem. (d) Write down the Lagrangean and derive the first-order necessary conditions. (e) Does there exist a solution x to the first-order necessary conditions with x1=0 ? (f) Does there exist a solution x to the first-order necessary conditions with x2=0 ? (g) Derive the consumer's demand correspondence x(p,w)