Determine the shadow price corresponding to the solution of the following constrained extremum problem.
  • Maximize $z(x,y)=xy$
  • Subject to $5x+y=25$
  • Where $x,y\geq0$
$\lambda=2\frac{1}{2}$
$\lambda=-2\frac{1}{2}$
$\lambda=12\frac{1}{2}$
$\lambda=-12\frac{1}{2}$
Determine the shadow price corresponding to the solution of the following constrained extremum problem.
  • Maximize $z(x,y)=xy$
  • Subject to $5x+y=25$
  • Where $x,y\geq0$
Antwoord 1 correct
Correct
Antwoord 2 optie
$\lambda=-2\frac{1}{2}$
Antwoord 2 correct
Fout
Antwoord 3 optie
$\lambda=12\frac{1}{2}$
Antwoord 3 correct
Fout
Antwoord 4 optie
$\lambda=-12\frac{1}{2}$
Antwoord 4 correct
Fout
Antwoord 1 optie
$\lambda=2\frac{1}{2}$
Antwoord 1 feedback
Correct: $L(x,y,\lambda)=xy-\lambda(5x+y-25)$. We differentiate with respect to the variables $x$, $y$ and $\lambda$ and put the derivatives equal to zero:
  • $L'_x(x,y,\lambda)=y-5\lambda=0$,
  • $L'_y(x,y,\lambda)=x-\lambda=0$,
  • $L'_{\lambda}(x,y,\lambda)=-5x-y+25=0$.
From $L'_x(x,y,\lambda)=y-5\lambda=0$ it follows that $y=5\lambda$ and from $L'_y(x,y,\lambda)=x-\lambda=0$ it follows that $x=\lambda$. We plug this into $L'_{\lambda}(x,y,\lambda)=-5x-y+25=0$ and that gives $\lambda=2\frac{1}{2}$. Hence, $x=2\frac{1}{2}$ and $y=12\frac{1}{2}$. $z(2\frac{1}{2},12\frac{1}{2})=31\frac{1}{4}$. The boundaries result in $z(0,25)=0$ and $z(5,0)=0$. Consequently, $z(2\frac{1}{2},12\frac{1}{2})=31\frac{1}{4}$ is the maximum with shadow price $\lambda=2\frac{1}{2}$.

Go on.
Antwoord 2 feedback
Wrong: The Lagrange function is $L(x,y,\lambda)=xy-\lambda(5x+y-25)$.

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Antwoord 3 feedback
Wrong: $y=5\lambda$.

Try again.
Antwoord 4 feedback
Wrong: $y=5\lambda$.

Try again.