Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Functions of Several Variables
Let f : R2 → R be defined by
\(\rm{ f(x, y) = { \left\{ \begin{matrix} \dfrac{x^2y}{x^4 + y^2} & if (x, y) \ne (0, 0) \\\ 0 & if (x, y) = (0, 0) \end{matrix} \right.}}\)
Which of the following statements holds regarding the continuity and the existence of partial derivatives of f at (0, 0)?
1
Both partial derivatives of f exist at (0, 0) and f is not continuous at (0, 0)
2
Both partial derivatives of f exist at (0, 0) and f is continuous at (0, 0)
3
One partial derivative of f does NOT exist at (0, 0) and f is continuous at (0, 0)
4
One partial derivative of f does NOT exist at (0, 0) and fis NOT continuous at (0, 0)