ECO 3145: Spring/Summer 2022 Quiz 1
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ECO 3145: Spring/Summer 2022
Quiz 1
I) For each question, choose the correct answer.
1. Let f be a univariate function defined by f (x) = 100 + e一3x + 4x. f has a:
a) local minimum b) unique global minimum
c) unique global maximum d) no correct answer
2. Consider the matrix A = 1 1 5 3 1 A is
I+2 0 4J.
a) positive definite b) negative semi-definite
c) positive semi-definite d) negative define
3. Let p=(p1 , p2 , ..., pn ) be a price vector and m e R++ an income level. The budget set B(p, m) defined by B(p, m) = (x e Rn /p.x = m}
/ 0 +4 1\
is not a convex set. True or false?
4. Convexity is a sufficient condition for a point to be a minimum. True or false?
/ 3 1 +2\ /x\ /+1\
II) Consider the equation AX = B where A = 1+4 3 1 1 , X = 1y 1 and C = 1 5 1
I 2 +2 3 J IzJ I 7 J.
1. Is A singular or non-singular? justify your answer.
2. How many solutions does this equation have? justify your answer.
3. Solve for the solutions if they exist.
III) Consider the bivariate function f defined by f (x, y) = +3xy+x ln(x2 y).
1. Compute f (+2, ), f (1, 1), and deduce that f (1, 1) cannot be a global minimum.
2. Determine the partial derivatives f1 (x, y) = and f2 (x, y) = .
3. Can (1, 1) be a local minimum? justify your answer.
2022-06-10