Mathematics for Economists, 2023
Assignment 2, Marks : 50
1. Find the equation of the parabola y = ax2 +bx+c that passes through the
points (1, −3), (0, −6), (3, 15). (Hint: Determine a, b, and c. [5 marks]
2. A person is given a rope of length L with which to enclose a rectangular
area.
(a) If one of the sides is x, show that the area of the enclosure is A(x) =
Lx
− x2 , where 0 ≤ x ≤ L/2. Find x such that the area of the
2
rectangle is maximized (without using differentiation).
(b) Will a circle of circumference L enclose an area that is larger than
the one we found in part (a)?
[4+3 marks]
3. Suppose a1 , a2 , ..., an and b1 , b2 , ...bn be arbitrary real numbers. Prove the
Cauchy-Schwarz inequality:
(a1 b1 + a2 b2 + ... + an bn )2 ≤ (a21 + a22 + ... + a2n )(b21 + b22 + ... + b2n )
(1)
Hint: Define f (x) = (a1 x+b1 )2 +...+(an x+bn )2 . We can see that f (x) ≥
0. Write f (x) as Ax2 + Bx + C, where the expressions for A, B and C are
related to the terms in Equation (1) above. Because Ax2 + Bx + C ≥ 0 for
all x, we must have B 2 − 4AC ≤ 0. Complete your answer by providing
proper arguments and reasoning. [10 marks]
4. Find the following limits:
(x + h)3 − x3
, h 6= 0.
h
n
x −1
b. limx→1
, where n is a natural number.
x−1
x−1
c. limx→0+ √
x
a. limx→0
d. limx→∞
(ax − b)2
(a − x)(b − x)
[4 marks each=16]
5. Consider the following functions:
ax
, where a, b, c are positive real numbers.
+c
3
(b) f (x) = x3 − x2 − 6x + 1.
2
(a) f (x) =
bx2
1
Find the points at which f 0 (x) = 0 and also specify if the functions are
continuous at all points in the domain or not. Use formal arguments.
[6+6 marks]
2