Question

Consider the function \(f:\mathbb{R}^2\to\mathbb{R}\) defined by
\[ f(x,y)= \begin{cases} \dfrac{x^3+y^3}{\sqrt{x^2+2y^2}} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0). \end{cases} \]
Which of the following statements is/are correct?

• A. \(f\) is continuous at \((0,0)\)
• B. Partial derivatives \(f_x\) and \(f_y\) exist at \((0,0)\) and \(f_x(0,0)=0\), \(f_y(0,0)=0\)
• C. \(f\) is not differentiable at \((0,0)\)
• D. \(f\) is differentiable

Hint:

Bound |f(x,y)| by x^2 + y^2/sqrt(2) using sqrt(x^2+2y^2) >= |x| and >= sqrt(2)|y|; this single bound settles continuity, the partial derivatives, and differentiability at once.

Answer 1

The Correct Option is A, B, D

Step 1: Polar form.
$f(x,y)=r^2\cdot\frac{\cos^3\theta+\sin^3\theta}{\sqrt{\cos^2\theta+2\sin^2\theta}}$, bounded by $Mr^2$.

Step 2: Continuity.
$|f|\le Mr^2\to0$. TRUE.

Step 3: Partials.
Along axes both give 0. TRUE.

Step 4: Differentiability.
$|f(h,k)|/r\le Mr\to0$ uniformly in direction. Differentiable. (C) FALSE.

Step 5: Everywhere.
Smooth quotient elsewhere. TRUE.
\[ \boxed{\text{A, B, D}} \]