数学问题一问

5
Since x1^2 and x2^2 are convex
-x1^2 and -x2^2 are concave
So 1-x1^2-x2^2 is concave
6
This is the Cobb-Douglas function, I do the general form here.

Consider the Cobb-Douglas function, defined by f (K, L) = AKaLb. The Hessian of this function is










圖片參考：http://www.economics.utoronto.ca/osborne/symbols/lparen2.gif

a(a−1)AKa−2Lb
abAKa−1Lb−1

圖片參考：http://www.economics.utoronto.ca/osborne/symbols/rparen2.gif


abAKa−1Lb−1
b(b−1)AKaLb−2
Thus in order that f be concave we need a(a−1)AKa−2Lb ≤ 0, b(b−1)AKaLb−2 ≤ 0, and abA2K2a−2L2b−2(1 − (a + b)) ≥ 0. Thus f is concave if A ≥ 0, a ≥ 0, b ≥ 0, and a + b ≤ 1, and is strictly concave if A > 0, a > 0, b > 0, and a + b < 1.
7
Let P (3, 4,-3), Q (5,2,1) and R (2,-1,4).
PQ=(2,-2,4) ; QR=(-3,-3,3)
PQxQR
=(6,-18,-12)
equation of the plane in R3 passing through (3, 4,-3), (5,2,1) and (2,-1,4) is
6(x-3)-18(y-4)-12(z+3)=0
6x-18y-12z+18=0
x-3y-2z+3=0
8
I don't know what is the question want to ask. I just try to answer.
Let the product we buy is X1,X2,X3
Then budget constraint is
X1+2X2+3X3<=2+2+15=19