Given f: X-R, define the bordered Hessian matrix B,(a) 0f1 fn f af(ar) af(a) f1 f1 B(a) fn f Theorem 2.6. Given convex X CR, twice differentiable f: X-R, and the principal b(a),., bn+1() of B(a), For XCR, f is quasi-convex=bk(x)≤0,yx∈X,k. 2. For X CR4, f is quasi-concave -(1)b()<0,VTEX, v k 3. For X=Rn or Rn, bk()<0,V CEX, Vk22- f is strictly quasi-convex 4. For X= R or Rn,(-1) bk()<0,VaE X, vk22=f is strictly quasI-Concave.■ Example 2.8. For f(a, y)=ra+y, defined on R2+, where a, B>0, quasI-concave, if0≤a,B≤1; f is strictly quasi-concave,if0<a,B≤1anda≠1or3≠1 Example 2.9. For Cobb-Douglas function f(a, y)=x@y, defined on R2+,where B≥0, quasI-concave, ifa,β≥0; ve, if a,B>0Given f : X → R, define the bordered Hessian matrix Bf (x) : fi ≡ ∂f(x) ∂xi , fij ≡ ∂2f(x) ∂xi∂xj , Bf (x) ≡ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0 f1 ··· fn f1 f11 ··· f1n . . . . . . . . . fn fn1 ··· fnn ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . Theorem 2.6. Given convex X ⊂ Rn, twice differentiable f : X → R, and the principal minors b1(x),...,bn+1(x) of Bf (x), 1. For X ⊂ Rn +, f is quasi-convex =⇒ bk(x) ≤ 0, ∀ x ∈ X, ∀ k. 2. For X ⊂ Rn +, f is quasi-concave =⇒ (−1)kbk(x) ≤ 0, ∀ x ∈ X, ∀ k. 3. For X = Rn + or Rn, bk(x) < 0, ∀ x ∈ X, ∀ k ≥ 2 =⇒ f is strictly quasi-convex. 4. For X = Rn + or Rn, (−1)kbk(x) < 0, ∀ x ∈ X, ∀ k ≥ 2 =⇒ f is strictly quasi-concave.  Example 2.8. For f(x, y) = xα + yβ, defined on R2 ++, where α, β ≥ 0, f is ⎧ ⎪⎨ ⎪⎩ quasi-concave, if 0 ≤ α, β ≤ 1; strictly quasi-concave, if 0 < α, β ≤ 1 and α 9= 1 or β 9= 1.  Example 2.9. For Cobb-Douglas function f(x, y) = xαyβ, defined on R2 ++, where α, β ≥ 0, f is ⎧ ⎪⎨ ⎪⎩ quasi-concave, if α, β ≥ 0; strictly quasi-concave, if α, β > 0.  2—6