My Math Forum Multivariable Functions: Relative Extrema

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 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=_______.
 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?
 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)
 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
 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!

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