Discussions (1)What is the intuition for the FOC? (2) What is the intuition for the SOC? (3How to verify the SOC? (4) Is the SOC related to quasi-concavity? Example 2. 13. Verify the SoC for Example 2. 12. Example 2. 14. verify the SoC for Example 2. 12. Theorem 2.11.(Kuhn-Tucker). For differentiable f: R-R and G: Rn- Rm consider problem st.G(x)≥0. Let C(, A)=f(a)+AG(a). If a* is a solution and Dg; (a),i=l k corresponding to binding constraints are linearly independent, then there exists AER+ such that FOC:D2L(x*,)=0, Kuhn-Tucker condition: A G(r)=0. Theorem 2.12. Suppose f, gi, hi: Rn- R are C functions, i= l,..., m,j= 1,., k, and k< n. Let G=(91,., gm) and H =(h1, .. hk). If a* is a solution of the following problem V≡maxf(x) t.G(x)≥0, H(x)=0, and the vectors Dg;(a")and Dh; (a), i=l,., m, j=1, ...,k, corresponding to binding constraints are linearly independent, then there exists a unique A E Rn and ∈ RK such that (a)Df(x*)+入.G(x)+p·H(x2)=0; (b)入.G(x)=0.■Discussions: (1) What is the intuition for the FOC? (2) What is the intuition for the SOC? (3) How to verify the SOC? (4) Is the SOC related to quasi-concavity? Example 2.13. Verify the SOC for Example 2.12.  Example 2.14. Verify the SOC for Example 2.12.  Theorem 2.11. (Kuhn-Tucker). For differentiable f : Rn → R and G : Rn → Rm, consider problem V ≡ max x∈Rn f(x) (2.1) s.t. G(x) ≥ 0. Let L(x, λ) ≡ f(x)+λ·G(x). If x∗ is a solution and Dgj (x∗), i = 1, . . . , m, j = 1,..., k, corresponding to binding constraints are linearly independent, then there exists λ ∈ Rm + such that FOC: DxL(x∗ , λ)=0, Kuhn-Tucker condition: λ · G(x∗ )=0.  Theorem 2.12. Suppose f, gi, hj : Rn → R are C1 functions, i = 1, . . . , m, j = 1,..., k, and k < n. Let G ≡ (g1,...,gm)T and H ≡ (h1,...,hk)T . If x∗ is a solution of the following problem V ≡ max x∈Rn f(x) (2.2) s.t. G(x) ≥ 0, H(x)=0, and the vectors Dgi(x∗) and Dhj (x∗), i = 1, . . . , m, j = 1,..., k, corresponding to binding constraints are linearly independent, then there exists a unique λ ∈ Rm + and μ ∈ Rk such that (a) Df(x∗ ) + λ · G(x∗ ) + μ · H(x∗ ) = 0; (b) λ · G(x∗ )=0.  2—9