Midterm Thursday, June 7
Problem 1. [10 pt]
(a) Let f (x) be a one variable function. Explain using Taylor’s formula why f 0 (x0 ) = 0 and f 00 (x0 ) > 0
implies that f has a local minimum at the point x0 . [3 pt]
(b) Let g(x, y, z) be a function of three variables, which has a local minimum at (0, 0, 0).
Explain why we have
gx (0, 0, 0) + gy (0, 0, 0) + gz (0, 0, 0) = 0,
and why the inequality
gyy (0, 0, 0) ≥ 0
holds. [4 pt]
(c) What is the Taylor approximation of order 2 of the function f (x, y) = (1 + x ln(y))3 around the
point (0, 1) ? [3 pt]
Problem 2. [10 pt]
(a) Consider the set of points (x, y) such that x ≥ 0, y ≥ 0 and 2x + y ≤ 4. Is it open, closed,
bounded ? [2 pt]
Are the following assertions true or false ? Give a counterexample or a short argument depending
on your answer.
(b) A continuous function defined on a bounded set always has a global minimum. [2 pt]
(c) A continuous function defined on a closed set always has a global minimum. [2 pt]
(d) A function can have several local minimum points. [2 pt]
(e) A function can have several global minimum points. [2 pt]
Problem 3. [10 pt]
(a) For what values of α is the following matrix negative definite ? [4 pt]


−2 0 0
 0 −2 1 
0
1 α
(b) Write the matrix corresponding to the quadratic form
q(x, y, z) = 3x2 − 2xy − 2yz + y 2 + 2z 2
1
and discuss its definiteness (i.e. is it positive definite, positive semidefinite, indefinite, negative definite, ... ?). [3 pt]
(c) Suppose you have a function of two variables f (x, y) such that fx (0, 0) = fy (0, 0) = 0 and
det Hf (0, 0) < 0. What can you say about the behaviour of f around the point (0, 0) ? [3 pt]
Problem 4. [10 pt]
Consider the function of two variables
f (x, y) = ex + (y + x)2 + αx + βy
(a) Discuss, depending on the values of the α and β if f admits a local minimum point. [6 pt]
(b) Is it a global minimum ? [4 pt]
Problem 5. [10 pt]
Consider the function of two variables
x2 + y 2
)
2
(a) Find its local extremum points. Are they local minima or maxima ? [5 pt]
f (x, y) = x exp(−
(b) Discuss globality. [5 pt]
Hint : you can try to show and use the fact that when x2 + y 2 is big, the values taken by f (x, y) are
close to 0.
2