ES10005. Questions for discussion.
Exercises on Topics T3 and T4:
Topic 3:
Optimization problems and terminology. Finding critical
points. Optimization of functions of 2 variables.
Topic 4:
Derivatives and shapes of functions, concave and
convex functions, second-order conditions.
Q1. Use the unconstrained optimization problem
min f ( x ) , where f ( x)  ax 2  x
x
to show off your knowledge of the following terms:
objective function
parameter
first order condition
stationary point
local minimum
second order condition
strictly convex function
choice variable
(first) derivative function
critical point
second derivative function
global minimum
convex function
unique global minimum
Q2. Solve:
min f ( x , y ) , where f ( x , y )  5x 2  6 xy  2 y 2  2 x  2 y  1000
x,y
Q3. Discuss:
max f ( x ) , where f ( x )  cos x .
x
(Start with a graph of the function, and confirm your comments
about the solution by treating the problem algebraically.)
ES10005 Classes 3&4
-1-
Q4. Solve instantly and without writing anything:
min ( p  1) 2  (q  2) 2  (r  3) 2  ( s  4) 2  (t  5) 2  (u  17) 2
p ,q ,r , s ,t ,u
Now algebraically confirm your brilliant flash of insight. (No need to
discuss SOC.)
Q5. If I charge $ P per person for admission to my next rock concert, the
audience will be of size A( P) .
(a)
(b)
(c)
(d)
(e)
(f)
Sketch A( P) .
Interpret A(0) .
What is the value of A() ?
Discuss the likely signs of the functions A( P), A ( P), A ( P)
when P  0 .
The costs of putting on the concert consist of a fixed cost $C ,
plus an additional $k for every person who attends. If my aim
is to maximize the profit from the concert, what is the
objective function? Hint: profit=revenue-cost.
In fact A( P)  10000e 0.025 P and k  10 . What is the best price to
charge?
Q6. Convex, concave, both or neither?
(a)
(d)
(g)
f ( x)  0
f ( x)  10x 2
f ( x)  exp( x)
f ( x )  ln x
(i)
(j)
(k)
(l)
ES10005 Classes 3&4
f ( x)  x
f ( x)  1 / x
f ( x)  x 0.9
(b)
(e)
(h)
f ( x)  100 (c)
f ( x)  x 3
(f)
f ( x)  exp( x)
( x  0)
( x  0)
( x  0)
( x  0)
-2-
f ( x)  10x
f ( x)  x 4