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ASSIGNMENT 5
BEEQ 1013 INTRODUCTION TO MATHEMATICAL ECONOMICS

BEEQ 1013 INTRODUCTION TO MATHEMATICAL ECONOMICS

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NUR LIYANA BINTI YAHAYA (255863)
Bachelor of Science with Honours (Business

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a)
Determine where the function f(x) =
is concave downward.
= x3 — 6x2 + 18x — 15
= 3x2 -12x+ 18
6x - 12
Let = O
6x — 12 = o
6x = 12
x3 —6x2 + 18x
— 15 is concave upward and where it
Interval
Test
1
3
f"(x)
—6
6
Sign of f"(x)
f (x) will concave downward on the interval ( —co , 2 ) and concave upward on the
interval (2 ,

= 3x2 -12x+ 18
= 6x - 12
Let f "(x) = 0
6x — 12 = o
6x = 12
f (x) will concave downward

f (x) will concave downward on the interval ( —co , 2 ) and concave upward on the
interval (2 ,

= x3 — 6x2 + 18x — 1
= 3x2 -12x + 18
= 6x - 12
Let f "(x) = 0
6x — 12 = o
6x = 12
co,2),

a)
Determine where the function f(x) =
is concave downward.
x3 -6x2 + 18x
— 15 is concave upward and where it

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b) Suppose that the cost function for a manufacturer is given by C(x) = O. 000001x3 —
O. 003x2 + 5x + 1000. Describe the behavior of the marginal cost function.
i) Find the marginal cost function.
Marginal cost function, C' (x)
c'(x) = (0.000001+ - 0.003x
0.000003x2 — 0.006x + 5
2 + 5x + 1000)

Marginal cost function, C' (x)
— (0.000001+ - 0.003x2 + 5x + 1000)
0.000003x2 — 0.006x + 5

b) Suppose that the cost function for a manufacturer is given by C(x) = O. 000001x3 —
O. 003x2 + 5x + 1000. Describe the behavior of the marginal cost function.

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b) Suppose that the cost function for a manufacturer is given by C(x) = 0.000001x3 —
0.003x2 + 5x + 1000. Describe the behavior of the marginal cost function.
ii) Describe the behavior of the marginal cost function whether concave upward or concave downward,
prove your answer.
c'(x) = 0.000003+ - 0.006x + 5
= 0.000006x
— 0.006
Let = O
0.000006x - 0.006 = o
0.000006x = 0.006
x = 1000
( —m, 1000), ( 1000 ,+m)
Interval
( on, 1000)
( 1000 , +00)
Test
200
1200
Sign of C"(x)
-0.0048
0.0012
Marginal cost function will concave downward on the interval ( —cn , 1000 ) and concave
upward on the interval ( 1000 , +00

Marginal cost function will concave downward on the interval ( —co , 1000 ) and concave
upward on the interval ( 1000 , +00

0001 ) ' (0001 )
0001 = x
900'0 = X90000(YO
o = - X900000'O
90(YO - x900000 0
S + X900 0
• -zXEOOOOOO

( 00, 1000)
( 1000 , +00)
200
1200
-0.0048
0.0012

ii) Describe the behavior of the marginal cost function whether concave upward or concave downward,
prove your answer.

b) Suppose that the cost function for a manufacturer is given by C(x) = 0.000001x3 —
0.003x2 + 5x + 1000. Describe the behavior of the marginal cost function.
) Describe the behavior of the marginal cost function whether concave upward or concave downward,

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c)
A company has establish that the revenue function in Ringgit Malaysia is R(x) = 2x3 +
40x + 8x and the cost function in Ringgit Malaysia is C(x) = 3x3 + 19x2 + 80x 800.
i) Find the level of production that will yield a maximum profit for the company
R(x) = 2x3 + 40x2 + 8x
C (X) = 3X3 + 19X2 + 80X — 800
Profit P(x) = P(x) - C(x)
= 2x3 + 40x2 + 8x - (3x3 + 19x2 + 801-800)
= 2x3 + 40x2 + 8x - 3x3 - 19x2 - 80x + 800
= -x3 + 21x2 — 72x + 800

Profit P (x)
= 2X3 + 40X2 + 8X
= 2X3 + 40X2 + 8X
- (3x3 + 19x2 + 801-800)
- 3x3 - 19x2 - 80x + 800
= -x3 + 21x2 — 72x + 800

c) A company has establish that the revenue function in Ringgit Malaysia is R(x) = 2x3 +
40x + 8x and the cost function in Ringgit Malaysia is C(x) = 3x3 + 19x2 + 80x — 800.

i) Find the level of production that will yield a maximum profit for the company

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c)
A company has establish that the revenue function in Ringgit Malaysia is R(x) = 2x3 +
40x + 8x and the cost function in Ringgit Malaysia is C(x) = 3x3 + 19x2 + 80x
800.
Differentiate profit function,
= —(-x3 + 21x2 - 72x + 800)
P'(x) = —3x2 + 42x — 72
Let P'(x) = O
-3x2 + 42x - 72 = o
3x2 - 42x + 72 = o
(x - - 12) = o
= —6X + 42
Let -
— — (12) 42
= —30 < 0 (maximum profit)
Let = +42
= 30 > o
The level of production that will yield a maximum profit is x = 12
(minimum profit)

Differentiate profit function,
= —(-x3 + 21x2 - 72x + 800)
P'(x) = —3x2 + 42x — 72
Let P'(x) = O
-3x2 + 42x -72 = o
3x2 -42x + 72 = o
(x - - 12) = o

c) A company has establish that the revenue function in Ringgit Malaysia is R(x) = 2x3 +
40x + 8x and the cost function in Ringgit Malaysia is C(x) = 3x3 + 19x2 + 80x — 800.
Differentiate profit function,

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c)
= 2x3 +
A company has establish that the revenue function in Ringgit Malaysia is R(x)
40x + 8x and the cost function in Ringgit Malaysia is C(x) = 3x3 + 19x2 + 80x
- 800.
ii) Find the price per unit to maximize the profit
Maximum profit production (x) = 12
Substitude (x) = 12 into profit function
—x3 + 21x2 - 72x + 800
-(12)3 + — 72(12) + 800
= 1232
Price per unit to maximize the profit is RM1232

Maximum protit production (x) = 1 •
Substitude (x) = 12 into profit function
—x3 + 21x2 - 72x + 800
-(12)3 + — 72(12) + 800
= 1232
Price per unit to maximize the profit is

Maximum profit production (x) = 12
Substitude (x) = 12 into profit function
P(x) = —x3 + 21x2 - 72x + 800
-(12)3 + — 72(12) + 800
= 1232

Maximum profit production (x) = 12
Substitude (x) = 12 into profit function
P(x) = —x3 + 21x2 - 72x + 800
-(12)3 + — 72(12) +
= 1232
Price per unit to maximize the profit i

ii) Find the price per unit to maximize the profit

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