Question

Let $f(x,y)=x^{3}-2y^{3}$. The curve along which $\nabla^{2} f = 0$ is
 

• A. $x=\frac{\sqrt{2}}{2}y$
• B. $x=2y$
• C. $x=\sqrt{6}y$
• D. $x=-\frac{y}{2}$

Hint:

For polynomial functions, the Laplacian simplifies quickly—differentiate twice and sum the results.

Answer 1

The Correct Option is B

Step 1: Compute the Laplacian. 
$f_x = 3x^{2}, f_{xx}=6x$ 
$f_y = -6y^{2}, f_{yy}=-12y$ 
Thus, $\nabla^{2} f = f_{xx} + f_{yy} = 6x - 12y$. 
 

Step 2: Set the Laplacian equal to zero. 
$6x - 12y = 0 $$\Rightarrow$ $x = 2y$. 
 

Step 3: Conclusion. 
The required curve is $x = 2y$.