Let $f(x,y)=x^{3}-2y^{3}$. The curve along which $\nabla^{2} f = 0$ is
Let $f(x,y)=x^{3}-2y^{3}$. The curve along which $\nabla^{2} f = 0$ is
Show Hint
- $x=\frac{\sqrt{2}}{2}y$
- $x=2y$
- $x=\sqrt{6}y$
- $x=-\frac{y}{2}$
The Correct Option is B
Solution and Explanation
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 $x = 2y$.
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