non linearfunctionsolution

7/26/2019 Non Linearfunctionsolution

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1. Determine whether solutions exist for each of the following quadratic

equations. Where they do find the solution(s).

Firstly determine whether solutions exist using the following criteria:

042

> acb Two solutions

042 = acb One solution04

2== acbtwo solutions exist

( ) 2

22

!2

42

2

42

=

=

=

a

acbbx

22

22=

+=x

02

22=

=x

(ii) ( ) ( ) 0!$% =+ xx

&ultiply out the 'uadratic

0$%% 2

= xx

(i)ide across by %

022

= xx

a!" b#!" c#2

( ) ( ) ( ) 0*2!4!4 22 >== acb two solutions exist

( ) 2%!

!2

*!

2

42

=

=

=

a

acbbx

22

%!=

+=x

!2

%!

=

=x

(iii) 0!$24* 2 =+ xx

a*" b#24" c!$

( ) ( ) ( ) 0+,$+,$!$*4244 22 === acb one solution

( ) %%-!!.

24

*2

024

2

42

==

=

=a

acbbx

(iv) 0%2% 2 =++ xx

a%" b2" c%

( ) ( ) ( ) 0%2%$4%%424 22


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(v) 02!!!2 2 =+ xx

a2" b!!" c#2!

( ) ( ) ( ) 02.*!$.!2!2!24!!4 22 >=+== acb two solutions

( ) 4!,!!

22

2.*!!

2

42

=

=

=a

acbb

x

+-!4

!,!!=

+=x ,

4

!,!!=

=x

/)i 0!02 2 =++ xx

a#2" b!" c!0

( ) ( ) ( ) 0.!!024!4 22 >== acb two solutions

( ) 4*!

22

.!!

2

42

=

=

=

a

acbbx

24

*!

=

+

=x +-24

*!

=

=x

2 A firms demand function for a good is given by P 1!"#$% and their total

cost function is given by &' $!!% .

i) *btain an ex+ression for total revenue +rofit in terms of %

Total 1e)enue -3

T1 /!0,#233 !0,3#232

rofit T1#T5

rofit !0,3#232#200#%3 #2326!043#200

ii) ,or what values of % does the firm brea- even

Firm brea7s e)en where rofit 0

#2326!043#200 0

a #2" b!04" c#200

iii) llustrate the answer to (ii) using s-etches of the total cost function/

the total revenue function and the +rofit function

roft !!+0


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Note: 8rea7 e)en where rofit 0 or T1T5-

iv) ,rom the gra+h estimate the maximum +rofit and the level of out+ut

for which +rofit is maximised

&aximum profit at max point on profit cur)e-

&ax profit !!+0 at 3 2$

%- What is the +rofit maximising level of out+ut for a firm with the marginal

cost function 0' 1.%$#12%! and a marginal revenue function 03

$4!#$!%5

rofit is maximised where &1&5

2.0#203 !-$32#!+36$0

!-$326+3#2200

a!-$" b+" c#220

( ) ( ) ( )

( )

%*-!%"2,-!0

2-%

.+-%,+

2-%

!40.2++

$-!2

220$-!4++ 2

==

=

+=

=

QQ

Q

rofit maximising le)el of output is 3 !0-2, /can9t ha)e negati)e output

4- &he demand function for a good is given as % 1!#1!P. ,ixed costs

associated with +roducing that good are 6! and each unit +roduced costs an

extra 67.i) *btain an ex+ression for total revenue and total costs in terms of %

rofit

T1

T5

#+00

0

+00

!000

!+00

2000

0 !0 20 %0 40 +0 $0

3

T5 T1 rofit

rofit

T1

T5

3 2$


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T1 -3

3 !%0#!0

!0 !%0#3

!%#3!0

T1 /!%#3!03 !%3#0-!32

T5 F56;5

T5 $0643

ii) ,or what values of % does the firm brea- even

Firm brea7s e)en where T1 T5

!%3#0-!32$0643

#0-!326*3#$00

a#0-!" b*" c#$0

( ) ( ) ( )

( )

,+-.2"2+-,

2-0

++-,*

2-0

24.!*

!-02

$0!-04** 2

==

=

=

=

QQ

Q

iii) *btain an ex+ression for +rofit in terms of % and s-etch its gra+h

iv) 8se the gra+h to confirm your answer to (ii) and to estimate maximum

+rofit and the level of out+ut for which +rofit is maximised

rofit T1#T5

rofit !%3#0-!32#$0#43#0-!326*3#$0

#!00

#+0

0

+0

!00

!+0

200

0 !0 20 %0 40 +0 $0 ,0 .0 *0

3

rofit

rofit

8rea7


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non linearfunctionsolution

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