My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By JeffM1
Reply
 
LinkBack Thread Tools Display Modes
August 8th, 2016, 04:49 PM   #1
Newbie
 
Joined: Aug 2016
From: United States

Posts: 3
Thanks: 0

Multivariable Functions: Relative Extrema

Hello, I have some problems with this question:

A production function P is given by P=f(m,k)=2.4m^2−0.1m^3+0.99k^2−0.06k^3
where l and k are the amounts of labor and capital, respectively, and P is the quantity of output produced. Find the values of m and k that maximize P.

Solution: To find the critical points we need to solve the system
Pm=_______ and Pk=________ .
The first equation gives that m=______ or m=_______. From the second equation we get k=______ or k=______.
This implies that there are four critical points (0,0), (0,__), (__,0), and (__,__).

At (0,__) we have that D(0,__)=_________ which is (pick one: >/=/<) zero. By the second-derivative test there is (relative max/relative min/no relative extrema) at (0,__).

At (__,0) we have that D(__,0)=_________ which is (pick one: >/=/<) zero. By the second-derivative test there is (relative max/relative min/no relative extrema) at (__,0).

Because D(__,__)= __________ (pick one: >/=/<) zero and Pmm (__,__)= ___________ (pick one: >/=/<) zero by the second-derivative test there is (relative max/relative min/no relative extrema) at this point.

The maximim output is obtained when m=______ and k=_______.
Jjgog is offline  
 
August 9th, 2016, 05:02 AM   #2
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,310
Thanks: 551

What is the relevance of L? Is it supposed to be in the production function? What is M? Are there constraints (there usually are constraints in an economics problem because in real life there are always constraints)? By constraints I mean in addition to the standard non-negativity constraints. (Do you know that if there are ANY constraints, mathematically they have to be explored separately?)

Just looking at the math

$P(m,\ k) = 2.4m^2 - 0.1m^3 + 0.99k^2 - 0.06k^3$

Can you find the partial derivatives or is that what you are asking?
JeffM1 is offline  
August 9th, 2016, 10:50 AM   #3
Newbie
 
Joined: Aug 2016
From: United States

Posts: 3
Thanks: 0

Sorry the original variable was "l" but I changed it to "m" so that it'd be easier to see, I forgot to change that one. I can solve for Pm and Pk, but when I enter critical points that I found (0,0), (0,11),(16,0)and(16,11) it tells me I'm wrong. (and I have no idea about the questions below)
Jjgog is offline  
August 9th, 2016, 02:00 PM   #4
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,310
Thanks: 551

I agree with you about the absurd ease of making mistakes when using minuscule l.

$P = 2.4m^2 - 0.1m^3 + 0.99k^2 - 0.06k^3 \implies$

$P_m = 4.8m - 0.3m^2 = m(4.8 - 0.3m)\ and\ P_k = k(1.98 - 0.18k) \implies$

$P_m = 0 = P_k\ at\ (0,\ 0),\ (16,\ 0),\ (0,\ 11),\ and\ (16,\ 11).$

Are you using some sort of computer program to enter your answers? It is unusual to to specify a point or a function as L, K rather than K, L. Try (11, 16).

The remaining part of the question relates to the second derivative test. Do you know it? It really is not technically applicable to (0, 0), which is a boundary point, but it works.
Thanks from Jjgog
JeffM1 is offline  
August 9th, 2016, 09:00 PM   #5
Newbie
 
Joined: Aug 2016
From: United States

Posts: 3
Thanks: 0

Looks like it was the computer being picky, I juggled the answers around and it ended up correct. I've also figured out the rest of the problem. Thanks!
Jjgog is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
extrema, functions, multivariable, relative



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Extrema of multivariable functions - systems of equations Akcope Calculus 5 March 8th, 2014 11:28 AM
Continuity of multivariable functions bigli Calculus 5 December 10th, 2009 08:08 PM
Using Maxima to find relative extrema adanedhel728 Applied Math 0 September 15th, 2009 06:20 PM
All extrema if they..... ArmiAldi Calculus 1 March 19th, 2008 01:46 AM
extrema mia6 Calculus 1 October 21st, 2007 05:55 PM





Copyright © 2019 My Math Forum. All rights reserved.