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  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization convex/concave functions
  5. Functions of two variables
  6. Second-order condition
  7. Exercise 2

Exercise 2

On which part of the domain is the function $z(x,y)=3x^2+2xy+y^3$ concave?
There is no region where the function is concave.
The function is concave in each region.
The function is concave where $x<0$ and $y<1$.
The function is concave where $x\geq 0$ and $y\geq 1$.
On which part of the domain is the function $z(x,y)=3x^2+2xy+y^3$ concave?
Antwoord 1 correct
Correct
Antwoord 2 optie
The function is concave in each region.
Antwoord 2 correct
Fout
Antwoord 3 optie
The function is concave where $x<0$ and $y<1$.
Antwoord 3 correct
Fout
Antwoord 4 optie
The function is concave where $x\geq 0$ and $y\geq 1$.
Antwoord 4 correct
Fout
Antwoord 1 optie
There is no region where the function is concave.
Antwoord 1 feedback
Correct: The second-order partial derivative $z''_{xx}(x,y)=6$ is positive for all values of $x$ and $y$. Hence, the function is concave in no region.

Go on.
Antwoord 2 feedback
Wrong: Note that the second-order partial derivative $z''_{xx}(x,y)=6$ is positive for all values of $x$ and $y$.

See Second-order condition.
Antwoord 3 feedback
Wrong: Note that the second-order partial derivative $z''_{xx}(x,y)=6$ is positive for all values of $x$ and $y$.

See Second-order condition.
Antwoord 4 feedback
Wrong: The function is convex in this region.

See Convex and concave.