Question

1. for the following constrained - optimization problems, do all of the calculations until you have set up a lagrange multiplier problem. write down the unknowns and the equations, but you do not need to solve them! (indeed, i believe that without a computer you are probably stuck...) (a) maximize (f(x,y)=e^{x^{2}y}) subject to (g(x,y)=x^{2}+y^{2}=1). (b) minimize (f(x,y,z)=sin(x^{2}+y^{4}+z^{6})) subject to (g(x,y,z)=3x^{2}+2y^{2}+z^{2}=7). (c) maximize (f(x,y,z)=ln(4 + x^{2}+sin(yz^{3}))) subject to (g(x,y,z)=x^{2}+y^{2}=49) and (h(x,y,z)=2x + 3y - z = 7).

Explanation:

Step1: Recall Lagrange - multiplier method

The Lagrange function for a function $f(x,y)$ subject to a constraint $g(x,y)=c$ is $L(x,y,\lambda)=f(x,y)-\lambda(g(x,y) - c)$.

Step2: For part (a)

The function is $f(x,y)=e^{x^{2}y}$ and the constraint is $g(x,y)=x^{2}+y^{2}=1$. The Lagrange function is $L(x,y,\lambda)=e^{x^{2}y}-\lambda(x^{2}+y^{2}-1)$.

Step3: For part (b)

The function is $f(x,y,z)=\sin(x^{2}+y^{4}+z^{6})$ and the constraint is $g(x,y,z)=3x^{2}+2y^{2}+z^{2}=7$. The Lagrange function is $L(x,y,z,\lambda)=\sin(x^{2}+y^{4}+z^{6})-\lambda(3x^{2}+2y^{2}+z^{2}-7)$.

Step4: For part (c)

The function is $f(x,y,z)=\ln(4 + x^{2}+\sin(yz^{3}))$ and the constraint is $g(x,y,z)=x^{2}+y^{2}=49$ and $h(x,y,z)=2x + 3y - z=7$. The Lagrange - function is $L(x,y,z,\lambda,\mu)=\ln(4 + x^{2}+\sin(yz^{3}))-\lambda(x^{2}+y^{2}-49)-\mu(2x + 3y - z - 7)$.

Answer:

(a) $L(x,y,\lambda)=e^{x^{2}y}-\lambda(x^{2}+y^{2}-1)$
(b) $L(x,y,z,\lambda)=\sin(x^{2}+y^{4}+z^{6})-\lambda(3x^{2}+2y^{2}+z^{2}-7)$
(c) $L(x,y,z,\lambda,\mu)=\ln(4 + x^{2}+\sin(yz^{3}))-\lambda(x^{2}+y^{2}-49)-\mu(2x + 3y - z - 7)$