Theorem 2.3. Given convex XCR, twice differentiable f: X-R, 1. f is convex D2f(x)≥0,x∈X 2. D f(a)>0,VIEX f is strictly convex 3. f is concave← (x)≤0,Vx∈X. 4. D f(a)<0, VaEX f is strictly concave Corollary 2.1. Given convex X C R, twice differentiable f: X -R, and let d(r,, ik,(a) be a k x k principal minors of D f(ar) 1. f is convex+dta…4}(x)≥0,Vx∈X,k,V{t1,…,i} 2. f is concave(-1)d1…xh(x)≥0,x∈X,Vk,V{i,…,i} 3. dk()>0,VaEX, k.= f is strictly convex 4.(-1)dk(x)>0,Vx∈X, f is strictly concave. I 3 xample25.“ f is strictly convex”#“D2f>0.E.g.,f(x)=x4 Example 2.6. For f(a, y)=a+y, defined on R2 +, where a, B20, concave if0≤a,B≤1; f strictly concave, if 0<a, B<l. Example 2.7. For Cobb-Douglas function f(a, y)=y, defined on R2+,where Concave, ifa,B≥0,a+B≤1; strictly concave, if a,B>0, a+B<1. Proposition 2.3. Let f: X-R be differentiable. Then, (1) f is concave→Df(x)·(y-x)≥f(y)-f(x), for all a,y∈X (2) f is strictly concave←→Df(x)·(y-x)>f(y)-f(x), for all x,y∈X,x≠yTheorem 2.3. Given convex X ⊂ Rn, twice differentiable f : X → R, 1. f is convex ⇐⇒ D2f(x) ≥ 0, ∀ x ∈ X. 2. D2f(x) > 0, ∀ x ∈ X =⇒ f is strictly convex. 3. f is concave ⇐⇒ D2f(x) ≤ 0, ∀ x ∈ X. 4. D2f(x) < 0, ∀ x ∈ X =⇒ f is strictly concave.  Corollary 2.1. Given convex X ⊂ Rn, twice differentiable f : X → R, and let d{i1,··· ,ik}(x) be a k × k principal minors of D2f(x), 1. f is convex ⇐⇒ d{i1,··· ,ik}(x) ≥ 0, ∀ x ∈ X, ∀ k, ∀{i1, ··· , ik}. 2. f is concave ⇐⇒ (−1)kd{i1,··· ,ik}(x) ≥ 0, ∀ x ∈ X, ∀ k, ∀{i1, ··· , ik}. 3. dk(x) > 0, ∀ x ∈ X, ∀ k. =⇒ f is strictly convex. 4. (−1)kdk(x) > 0, ∀ x ∈ X, ∀ k. =⇒ f is strictly concave.  Example 2.5. “ f is strictly convex” > “ D2f > 0”. E.g., f(x) = x4.  Example 2.6. For f(x, y) = xα + yβ, defined on R2 ++, where α, β ≥ 0, f is ⎧ ⎪⎨ ⎪⎩ concave, if 0 ≤ α, β ≤ 1; strictly concave, if 0 < α, β < 1.  Example 2.7. For Cobb-Douglas function f(x, y) = xαyβ, defined on R2 ++, where α, β ≥ 0, f is ⎧ ⎪⎨ ⎪⎩ concave, if α, β ≥ 0, α + β ≤ 1; strictly concave, if α, β > 0, α + β < 1.  Proposition 2.3. Let f : X → R be differentiable. Then, (1) f is concave ⇐⇒ Df(x) · (y − x) ≥ f(y) − f(x), for all x, y ∈ X. (2) f is strictly concave ⇐⇒ Df(x) · (y − x) > f(y) − f(x), for all x, y ∈ X, x 9= y.  2—4