Economics 214
Lecture 7
Introduction to Functions Cont.
Extreme Values




The largest value of a function over its entire
range is call its Global maximum.
The smallest value over the entire range is
called the Global minimum.
Largest value within a small interval is a
local maximum.
Smallest value within a small interval is
called a local minimum.
Figure 2.8 A Function with
Extreme Values
Average Rate of Change
The average rate of change of a function over some
interval is the ratio of the change of the value of the
dependent variable to the change in the value of the
independen t variable over that interval.
The average rate of change of the function y  f(x)
over the closed interval x A ,xB  is
Δy f  xB   f  x A 

Δx
xB  x A
Average Rate of Change
Linear Function
Let y    x
for the interval x A , xB , the average rate of
change is
Δy f  xB   f  x A    xB     x A 


Δx
xB  x A
xB  x A

 xB  x A 
xB  x A

This is the slope of our linear function.
Average Rate of Change for
Nonlinear function
Consider our function from lecture 6
2
y
1 x
Interval 1,2
y 2 1  2   2 1  1 2 3  1  1 3



 1 3
x
2 1
1
1
Interval 1,5
y 2 1  5  2 1  1 2 6  1  2 3



 1 6
x
5 1
4
4
For nonlinear function, average rate of change not
the same over all intervals.
Secant Line
Average rate of change of a function over some
interval can be deplicted using a secant line. A secant
line connects two points on the graph of a function
with a straight line. Any point (x',y') on the secant line
connecting the two points  x A , y A  and  xB , y B  will
satisfy th e equation
 y'-y A    f xB   f x A x' x A  or



xB  x A

 f xB   f x A 
y'  y A  
  x' x A 
x

x
B
A


Graph of Average Rate of
Change for Nonlinear Function
Average Rate of Change
1.2
1
0.8
Secant B
0.6
Secant A
y
0.4
0.2
0
0
1
2
3
x
4
5
6
Concavity and Convexity



The concavity of a univariate function is
reflect by the shape of it graph and its
secant line.
Function is strictly concave in the interval
[xA,xB] if the secant line drawn on that
interval lies wholly below the function.
Function is strictly convex in the interval
[xA,xB] if the secant line drawn on that
interval lies wholly above the function.
Figure 2.10 Strictly Concave
and Strictly Convex Functions
Strictly Concave
The function f(x) is strictly concave in an interval if, for
any two distinct points x A and x B in that interval, and for
all values of  in the open interval ( 0 ,1 ),


 
 
f x A  1   x B  f x A  1    f x B
Strictly Convex
The function f(x) is strictly concex in an interval if, for
any two distinct points x A and x B in that interval, and for
all values of  in the open interval ( 0 ,1 ),


 
 
f x A  1   x B  f x A  1    f x B
Convexity and secant line
for a given valu e of  in the interval ( 0 ,1 ), the argrument
of the function in the interval (x A ,x B ) is x'  λx A  ( 1-λ) x B ,
where x A  x'  x B . The value of the function at the point x' is


f  x'  f λx A  ( 1-λ) x B . With the definition of secant line
the value on the secant line at x' is y'
    
 f xB  f x A 
A
B
A
( y ' y )  
λx

(
1
-λ
)
x

x

B
A
x

x


A
   


 f xB  f x A 
B
A
B
A





1


x

x

1


f
x

f
x

B
A
x

x


since y A  f ( x A ), if we add f ( x A ) to each side, we get
 

 
y'  f x A  1    f x B

    
Our Example
Consider our function y 
2
over the interval ( 1,5 ).
1 x
Let   1 2 , so x'  3.
1 1 2
2  3 3
1 2
f x A  (1   ) xB   f 3  
2 3
This is true for all values of  in the interval (0,1), hence
f  x A   (1   ) f  x A   (1)    
1
2
our function is strictly convex over the interval (1,5).
Concave
The function f(x) is concave in an interval if, for
any two distinct points x A and x B in that interval, and for
all values of  in the open interval ( 0 ,1 ),


 
 
f x A  1   x B  f x A  1    f x B
Convex
The function f(x) is concex in an interval if, for
any two distinct points x A and x B in that interval, and for
all values of  in the open interval ( 0 ,1 ),


 
 
f x A  1   x B  f x A  1    f x B
Figure 2.11 Concave and
Convex Functions