Homework 3
Due Monday, November 8
Exercise 1. Show that if f is strictly quasi-concave, then it has a unique maximum x on
any interval [a, b]. Moreover, show that if f is differentiable at x with f ′(x) = 0, then x is
the unique maximum.
Exercise 2. Suppose f(x1, x2, x3) = x1x2 ln(x
2
1+x
2
2+x
2
3). Compute the directional derivative
at x = (1, 1, 1) in the direction (−4,−1, 1). Moreover, determine the direction of maximal
increase from the point (1, 1, 1).
Exercise 3. Show that if a function f : Rn → R has continuous second-order partial
derivatives {∂ijf}, then the cross partials are equal: ∂ijf = ∂jif .
Exercise 4. Carry out a detailed calculation showing that the directional second derivative
of f : Rn → R, in the direction v, is
d2
dh2
f(x+ hv)
∣∣
h=0
= vᵀHf (x)v
in which Hf (x) is the Hessian matrix at x.
Exercise 5. Consider the maximization problem
max
x1,x2
f(x1, x2, r) = max
x1,x2
−x21 − x1x2 − x
2
2 + 2rx1 + 2rx2
in which r is a parameter. Find the solution x∗(r) as a function of r. Writing f ∗(r) for the
maximal value, verify that
df ∗(r)
dr
=
∂f(x1, x2, r)
∂r
∣∣
x=x∗(r)
.
Exercise 6. Maximize ex1 + x2 + x3 subject to x1 + x2 + x3 = 1 and x
2
1 + x
2
2 + x
2
3 = 1.
Exercise 7. Show that a function f : [a, b] → R is Riemann integrable if and only if it is
bounded on [a, b] and for each � > 0, there exists a partition P of [a, b] such that the difference
between the corresponding upper and lower sums is less than �.
Exercise 8. Consider the algebra A generated by half open intervals (a, b] with −∞ ≤ a ≤
b ≤ ∞. Show that A consists exactly of all finite disjoint unions of half open intervals.
Exercise 9. Suppose (X,F) is a measurable space, and S /∈ F . Show that the σ-algebra
M(F ∪ {S}) generated by adding S to F consists of all sets of the form
(A ∩ S) ∪ (B ∩ (X \ S))
with A,B ∈ F .
Exercise 10. Find a countable collection of sets C that generates the Borel σ-algebra B(R).
Exercise 11. Given a set S ⊆ R, write S + x = {y ∈ R : y = s + x, s ∈ S} for the
translation of S by x. Suppose S ∈ B(R) is a Borel set, that λ is the Lebesgue measure, and
λ (S \ (S + x)) = 0 for every real number x. Prove that either λ(S) = 0 or λ(R \ S) = 0.
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