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Exercises on Optimization – I
Exercises I
Exercises II
1
Functions for two variables A function of two variables:
z = f ( x, y )
with partial derivatives
f
xy
fx =
∂ 2 f , f = ∂x∂y
∂f ∂f , fy = ∂y ∂x xx
∂ 2 f , f = ∂x 2
yy
∂ 2 f . = ∂y 2
Table: test for relative extremum: Condition First order
Maximum
Second order
Minimum
fx = f y = 0
fx = fy = 0
f xx , f yy < 0
f xx , f yy > 0
f xx f yy > f xy2
f xx f yy > f xy2
Exercise: III 1. A firm produces two different kinds A and B of a commodity. The daily cost of producing x units of A and y units of B is
C ( x, y ) = 0.04 x 2 + 0.01xy + 0.01y 2 + 4 x + 2 y + 500
Suppose that the firm sells all its output at a price per unit of 15 for A and 9 for B. Find the daily production levels x and y that maximize profit per day.
π ( x, y ) = 15 x + 9 y − (0.04 x 2 + 0.01xy + 0.01 y 2 + 4 x + 2 y + 500) Find the extreme values of the following functions and determine whether they are maxima or minima: i. z = x2 + xy + 2y2 +3 ii.
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