1. Home
  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization functions of two variables
  5. Stationary point
  6. Exercise 3

Exercise 3

Determine all the stationary points of $z(x,y)=\frac{1}{3}x^3-x+\frac{1}{3}y^3-4y$.
$(-1,-2)$, $(-1,2)$, $(1,-2)$ and $(1,2)$.
$(1,2)$
$(0,0)$
There are no stationary points.
Determine all the stationary points of $z(x,y)=\frac{1}{3}x^3-x+\frac{1}{3}y^3-4y$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$(1,2)$
Antwoord 2 correct
Fout
Antwoord 3 optie
$(0,0)$
Antwoord 3 correct
Fout
Antwoord 4 optie
There are no stationary points.
Antwoord 4 correct
Fout
Antwoord 1 optie
$(-1,-2)$, $(-1,2)$, $(1,-2)$ and $(1,2)$.
Antwoord 1 feedback
Correct:
  • $z'_x(x,y)=x^2-1$
  • $z'_y(x,y)=y^2-4$
$z'_x(x,y)=0$ gives $x=1$ or $x=-1$.
$z'_y(x,y)=0$ gives $y=2$ or $y=-2$.

This results in the four combinations.

Go on.
Antwoord 2 feedback
Wrong: Do not forget the negative solutions of a quadratic equation.

Try again.
Antwoord 3 feedback
Wrong: A stationary point is not a point where the function is equal to zero.

See Stationary point.
Antwoord 4 feedback
Wrong: What are the two partial derivatives?

Try again.