EC123 Notes
Topic 1: One-variable Optimisation: A function maps each element of a domain to a unique
element of its range or codomain. y is the value of the function or output.
An inequality is strict if replacing any “less than” and “greater than” signs with equal signs
never gives a true expression.
1.1: Convexity and concavity: Given that it is twice continuously differentiable, a function
f(x) is convex over D if and only if f '' ( x ) ≥ 0 for all x ∈ D ;concave over D if and only if f '' ( x ) ≤ 0
for all x ∈ D .
So, it follows that if a function is convex, its 1st derivative is increasing. And vice-versa.
A linear function is both convex and concave at the same time. y = ax + b, f’(x) = b, f’’(x) = 0
Example: Consider the function f(x) = ln(x) whose domain is X =R++¿¿ (all x which are +)
' 1 '' −1
f ( x )= >0 , f ( x ) = 2 <0
x x
Thus, the natural logarithm is increasing and concave.
Alternatively, A function f(x) is convex if for any two x1, x2 and any t ∈ [ 0,1 ] ,
tf ( x 1 ) + ( 1−t ) f ( x 2 ) ≥ f ( t x 1+ ( 1−t ) x 2 )
If the inequality holds as ≤ ,then the function is concave.
Equivalently: A function f is concave (convex) if the line segment that joins any two points
on the graph is below (above) the graph, or on the graph.
1.2: Extreme Points: Formally, c ∈ D is a maximum point if f ( x ) ≤ f ( c ) for all x ∈ D.
If c ∈ D is either a maximum or minimum point, then c is an extreme point or optimal point.
If the inequalities above are strict, then we refer to the extreme point as a strict maximum
or strict minimum.
Suppose the function f(x) is differentiable over an interval I and that c is an interior point of
I. Then c is an extreme point in I only if it is a stationary point, that is f’(c) = 0 (FOC)
Therefore, for the interior points of a differentiable function, a necessary condition for an
extreme point is that it is also a stationary point.
However, an extreme point does not imply a stationary point:
1) If a function is not differentiable over the whole domain
Minimum at d (but not differentiable here) and a maximum at b but neither are
stationary points (b is not an interior point)
2) A function has stationary points such as a local maximum/minimum or
inflection point, but these are not extreme points.
Thus f’(x) = 0 is not a sufficient condition to identify extreme points.
1.3: Existence of extreme points: If f is a continuous function over a closed bounded interval
[ x 0 , x 1 ] , then there exists a point c ∈[x 0 , x 1 ] where f has a minimum and a point
d ∈ [ x 0 , x 1 ] where f has a maximum so that f (c )≤ f ( x)≤ f ( d ) for all x ∈[ x 0 , x 1 ]
This is the extreme value theorem (EVT).
,If f is continuous over the closed bounded interval [ x 0 , x 1 ] and differentiable in the open
interval ( x 0 , x 1 ) ,then there exists at least one interior point c ∈ ( x 0 , x 1 ) such that
'
f ( x 1 ) −f ( x 0 )
f ( c )= This is the mean value theorem
x1 −x0
Rolle’s theorem: if a function is both continuous and differentiable over some interval
[x0, x1] where f(x0) = f(x1), there exists at least one stationary point.
1.4: Local extreme points: Sometimes it is useful to emphasize that a stationary point is a
maximum or a minimum but only over a given domain, rather than the whole domain.
So, a function f(x) has a local minimum/maximum at c if there exists an interval (a,b) such
that c ∈( a , b) and c is a maximum/minimum for the function defined over the interval (a,b).
Every extreme point is a local extreme point but not vice versa.
1.5 Inflection points: Points at which a function changes from being convex to being
concave, or vice versa, are called inflection points.
Inflection points are another way in which the FOC may fail to identify a maximum or
minimum, since the slope of the function may also equal 0 at such a point.
To identify an inflection point look at the second derivative and check whether the
signs flip below and above. If c is an inflection point, then f’’(c) = 0 and f’’(c) changes
sign for slightly smaller/bigger points on the domain. Or equivalently, the third order
derivative at c is non zero, since the slope here is at a maximum or minimum.
1.6: Sufficient conditions for extreme points: Suppose f(x) is a convex (concave) continuous
function over an interval I. If c is a stationary point for f in the interior of I, then c is a
minimum (maximum) point for f in I. (Sufficient but not necessary)
General conditions for stationary points:
Suppose the function f(x) has a stationary point x = c, that is f’(c) = 0.
Let f ( j) (x) be the jth derivative (provided it exists) and n the smallest number such that
(n )
f ( c ) ≠ 0.Then c is:
1) A local maximum if n is even and f (n )(c)<0
2) A local minimum if n is even and f (n )( c)>0
3) An inflection point if n is odd
Example: f(x) = −( x−1 )4 : f’(x) = -4(x – 1)3 = 0, therefore, x = 1
Differentiating again: f ' ' ( x )=−12 ( x −1 )2 , f ' ' (1 )=0
f (3 ) ( x )=−24 ( x −1 ) , f (3 ) ( 1 )=0
f (4 ) ( x ) =−24 , f 4 (1 )=−24
So the first nonzero derivative is the 4th, therefore n = 4 = even we see that f(4)(1) ¿ 0 .
Therefore, x = 1 is the value that gives a local maximum
Extreme points: The maximum and minimum of a differentiable univariate function f(x)
defined on a closed, bounded interval [a, b] can be obtained by identifying and comparing
all possible candidate points: any local maxima or local minima (by classifying all stationary
points), and the end points of the interval a and b
We can also classify a stationary point as an extreme point immediately if the specific
circumstances under which the FOC are sufficient to guarantee a max (or minimum) hold
, Topic 2: Logic and Useful Definitions:
2.1: Logic: A implies B, contraposition and equivalence:
A is necessary but not sufficient for B. We say: B holds only if A holds: B ⟹ A
A is neither necessary nor sufficient for B
A is sufficient but not necessary for B: We say B holds if A holds: A ⟹ B
A is both necessary and sufficient for B. We say: B holds if and only if A
holds (or: B holds iff A holds): A ⟺ B
The contrapositive principle: Notice that if A ⟹ B , this also suggests that not B implies not A
Every mathematical theorem can be formulated as one or more implications of the form
A ⟹ B with A being premises, assumptions and B being conclusions (deductively inferred).
A direct proof involves starting with the premises and working forward to the conclusions
An indirect or contrapositive proof exploits equivalence and proves implication by supposing
B is not true, and demonstrating that A must logically not be true either.
Note: deductive reasoning (logic) and inductive reasoning (inferred from observations)
Example: Show that f’’(x) = 0 is not sufficient for an inflection point
Consider the function: f(x) = x4
The derivatives are f’(x) = 4x3, f’’(x) = 12x2
Thus x = 0 yields f’’(x) = 0. However, f’’(x) ¿ 0 for all other x. Thus we need to check two
things: that the second derivative is zero (necessary condition) and that the function
changes from convex (concave) to concave (convex).
2.2: Continuity and Limits: Graphically, a function is continuous if its graph is connected and
has no jumps (your pen should not leave the paper while drawing).
A function f ( x ) is continuous at x=aif lim f ( x )=f ( a)
x→a
This can go wrong in the case where f(a) is not defined (0/0 for example) – the function is
not continuous everywhere -> use l’hopital’s rule
We can make f(x) as close as we like to A by making x sufficiently close to a, we say that
lim f ( x )= A . Or more formally:
x→ a
The function f(x) has a limit A as x tends to a, denoted by lim
x→ a
f ( x )= A , for each number ε > 0
there exists a number δ >0 such that |f ( x ) −A|< ε for every x with 0 ¿|x−a|<δ
With ε being the error in the measurement of the value at the limit and δ the distance to the
limit point. No matter how small ε is, we can find a δ that is small enough to be consistent
with the value of ε
One sided limits: It seems that the function has a limit at a if we take
−¿ ¿
values of x that are smaller than a. Mathematically, f ( x ) → A as x → a ,
where the minus in the superscript of a denotes that we are taking values
of x that come from the left (or below)
A necessary and sufficient condition for the limit at x = a to exist is that the
two one-sided limits of f at a exist and are equal.
2.3: L’Hopital’s Rule and Cobb-Douglas Functions:
Topic 1: One-variable Optimisation: A function maps each element of a domain to a unique
element of its range or codomain. y is the value of the function or output.
An inequality is strict if replacing any “less than” and “greater than” signs with equal signs
never gives a true expression.
1.1: Convexity and concavity: Given that it is twice continuously differentiable, a function
f(x) is convex over D if and only if f '' ( x ) ≥ 0 for all x ∈ D ;concave over D if and only if f '' ( x ) ≤ 0
for all x ∈ D .
So, it follows that if a function is convex, its 1st derivative is increasing. And vice-versa.
A linear function is both convex and concave at the same time. y = ax + b, f’(x) = b, f’’(x) = 0
Example: Consider the function f(x) = ln(x) whose domain is X =R++¿¿ (all x which are +)
' 1 '' −1
f ( x )= >0 , f ( x ) = 2 <0
x x
Thus, the natural logarithm is increasing and concave.
Alternatively, A function f(x) is convex if for any two x1, x2 and any t ∈ [ 0,1 ] ,
tf ( x 1 ) + ( 1−t ) f ( x 2 ) ≥ f ( t x 1+ ( 1−t ) x 2 )
If the inequality holds as ≤ ,then the function is concave.
Equivalently: A function f is concave (convex) if the line segment that joins any two points
on the graph is below (above) the graph, or on the graph.
1.2: Extreme Points: Formally, c ∈ D is a maximum point if f ( x ) ≤ f ( c ) for all x ∈ D.
If c ∈ D is either a maximum or minimum point, then c is an extreme point or optimal point.
If the inequalities above are strict, then we refer to the extreme point as a strict maximum
or strict minimum.
Suppose the function f(x) is differentiable over an interval I and that c is an interior point of
I. Then c is an extreme point in I only if it is a stationary point, that is f’(c) = 0 (FOC)
Therefore, for the interior points of a differentiable function, a necessary condition for an
extreme point is that it is also a stationary point.
However, an extreme point does not imply a stationary point:
1) If a function is not differentiable over the whole domain
Minimum at d (but not differentiable here) and a maximum at b but neither are
stationary points (b is not an interior point)
2) A function has stationary points such as a local maximum/minimum or
inflection point, but these are not extreme points.
Thus f’(x) = 0 is not a sufficient condition to identify extreme points.
1.3: Existence of extreme points: If f is a continuous function over a closed bounded interval
[ x 0 , x 1 ] , then there exists a point c ∈[x 0 , x 1 ] where f has a minimum and a point
d ∈ [ x 0 , x 1 ] where f has a maximum so that f (c )≤ f ( x)≤ f ( d ) for all x ∈[ x 0 , x 1 ]
This is the extreme value theorem (EVT).
,If f is continuous over the closed bounded interval [ x 0 , x 1 ] and differentiable in the open
interval ( x 0 , x 1 ) ,then there exists at least one interior point c ∈ ( x 0 , x 1 ) such that
'
f ( x 1 ) −f ( x 0 )
f ( c )= This is the mean value theorem
x1 −x0
Rolle’s theorem: if a function is both continuous and differentiable over some interval
[x0, x1] where f(x0) = f(x1), there exists at least one stationary point.
1.4: Local extreme points: Sometimes it is useful to emphasize that a stationary point is a
maximum or a minimum but only over a given domain, rather than the whole domain.
So, a function f(x) has a local minimum/maximum at c if there exists an interval (a,b) such
that c ∈( a , b) and c is a maximum/minimum for the function defined over the interval (a,b).
Every extreme point is a local extreme point but not vice versa.
1.5 Inflection points: Points at which a function changes from being convex to being
concave, or vice versa, are called inflection points.
Inflection points are another way in which the FOC may fail to identify a maximum or
minimum, since the slope of the function may also equal 0 at such a point.
To identify an inflection point look at the second derivative and check whether the
signs flip below and above. If c is an inflection point, then f’’(c) = 0 and f’’(c) changes
sign for slightly smaller/bigger points on the domain. Or equivalently, the third order
derivative at c is non zero, since the slope here is at a maximum or minimum.
1.6: Sufficient conditions for extreme points: Suppose f(x) is a convex (concave) continuous
function over an interval I. If c is a stationary point for f in the interior of I, then c is a
minimum (maximum) point for f in I. (Sufficient but not necessary)
General conditions for stationary points:
Suppose the function f(x) has a stationary point x = c, that is f’(c) = 0.
Let f ( j) (x) be the jth derivative (provided it exists) and n the smallest number such that
(n )
f ( c ) ≠ 0.Then c is:
1) A local maximum if n is even and f (n )(c)<0
2) A local minimum if n is even and f (n )( c)>0
3) An inflection point if n is odd
Example: f(x) = −( x−1 )4 : f’(x) = -4(x – 1)3 = 0, therefore, x = 1
Differentiating again: f ' ' ( x )=−12 ( x −1 )2 , f ' ' (1 )=0
f (3 ) ( x )=−24 ( x −1 ) , f (3 ) ( 1 )=0
f (4 ) ( x ) =−24 , f 4 (1 )=−24
So the first nonzero derivative is the 4th, therefore n = 4 = even we see that f(4)(1) ¿ 0 .
Therefore, x = 1 is the value that gives a local maximum
Extreme points: The maximum and minimum of a differentiable univariate function f(x)
defined on a closed, bounded interval [a, b] can be obtained by identifying and comparing
all possible candidate points: any local maxima or local minima (by classifying all stationary
points), and the end points of the interval a and b
We can also classify a stationary point as an extreme point immediately if the specific
circumstances under which the FOC are sufficient to guarantee a max (or minimum) hold
, Topic 2: Logic and Useful Definitions:
2.1: Logic: A implies B, contraposition and equivalence:
A is necessary but not sufficient for B. We say: B holds only if A holds: B ⟹ A
A is neither necessary nor sufficient for B
A is sufficient but not necessary for B: We say B holds if A holds: A ⟹ B
A is both necessary and sufficient for B. We say: B holds if and only if A
holds (or: B holds iff A holds): A ⟺ B
The contrapositive principle: Notice that if A ⟹ B , this also suggests that not B implies not A
Every mathematical theorem can be formulated as one or more implications of the form
A ⟹ B with A being premises, assumptions and B being conclusions (deductively inferred).
A direct proof involves starting with the premises and working forward to the conclusions
An indirect or contrapositive proof exploits equivalence and proves implication by supposing
B is not true, and demonstrating that A must logically not be true either.
Note: deductive reasoning (logic) and inductive reasoning (inferred from observations)
Example: Show that f’’(x) = 0 is not sufficient for an inflection point
Consider the function: f(x) = x4
The derivatives are f’(x) = 4x3, f’’(x) = 12x2
Thus x = 0 yields f’’(x) = 0. However, f’’(x) ¿ 0 for all other x. Thus we need to check two
things: that the second derivative is zero (necessary condition) and that the function
changes from convex (concave) to concave (convex).
2.2: Continuity and Limits: Graphically, a function is continuous if its graph is connected and
has no jumps (your pen should not leave the paper while drawing).
A function f ( x ) is continuous at x=aif lim f ( x )=f ( a)
x→a
This can go wrong in the case where f(a) is not defined (0/0 for example) – the function is
not continuous everywhere -> use l’hopital’s rule
We can make f(x) as close as we like to A by making x sufficiently close to a, we say that
lim f ( x )= A . Or more formally:
x→ a
The function f(x) has a limit A as x tends to a, denoted by lim
x→ a
f ( x )= A , for each number ε > 0
there exists a number δ >0 such that |f ( x ) −A|< ε for every x with 0 ¿|x−a|<δ
With ε being the error in the measurement of the value at the limit and δ the distance to the
limit point. No matter how small ε is, we can find a δ that is small enough to be consistent
with the value of ε
One sided limits: It seems that the function has a limit at a if we take
−¿ ¿
values of x that are smaller than a. Mathematically, f ( x ) → A as x → a ,
where the minus in the superscript of a denotes that we are taking values
of x that come from the left (or below)
A necessary and sufficient condition for the limit at x = a to exist is that the
two one-sided limits of f at a exist and are equal.
2.3: L’Hopital’s Rule and Cobb-Douglas Functions:
