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  3. Chapter 5: Optimization
  4. Optimization functions of two variables
  5. Second-order partial derivatives
  6. Exercise 3

Exercise 3

Consider the function $z(x,y)=\;^3\!\log (x^3y^4)$. Determine $z''_{xy}(x,y)$.
$z''_{xy}(x,y)=0$
$z''_{xy}(x,y)=\dfrac{3}{x \textrm{ln}(3)}$
$z''_{xy}(x,y)=\dfrac{4}{y\textrm{ln}(4)}$
$z''_{xy}(x,y)=\dfrac{-4}{x^3y^4\textrm{ln}(3)}$
Consider the function $z(x,y)=\;^3\!\log (x^3y^4)$. Determine $z''_{xy}(x,y)$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$z''_{xy}(x,y)=\dfrac{3}{x \textrm{ln}(3)}$
Antwoord 2 correct
Fout
Antwoord 3 optie
$z''_{xy}(x,y)=\dfrac{4}{y\textrm{ln}(4)}$
Antwoord 3 correct
Fout
Antwoord 4 optie
$z''_{xy}(x,y)=\dfrac{-4}{x^3y^4\textrm{ln}(3)}$
Antwoord 4 correct
Fout
Antwoord 1 optie
$z''_{xy}(x,y)=0$
Antwoord 1 feedback
Correct: $z'_x(x,y)=\dfrac{3x^2y^4}{x^3y^4\textrm{ln}(3)}=\dfrac{3}{x\textrm{ln}(3)}$. Hence, $z''_{xy}=z''_{yx}(x,y)=0$.

Go on.
Antwoord 2 feedback
Wrong: $z''_{xy}(x,y)\neq z'_x(x,y)$.

See Second-order partial derivatives.
Antwoord 3 feedback
Wrong: $z''_{xy}(x,y)\neq z'_y(x,y)$.

See Second-order partial derivatives.
Antwoord 4 feedback
Wrong: Do not forget the chain rule.

See Chain rule.