Given the Function $f$: $\mathbb{R^2} \rightarrow \mathbb{R}$ defined as the following: $f(x,y) = \left\{ \begin{array}{ll} 3x & \mbox{if } x = y \\ 0 & \mbox{if } x \ne y \end{array} \right., $ calculate the $\bigtriangledown f(0,0).$
I calculated this gradient and it gave the vector $(3,0)$ but in the solutions sheet is the vector $(0,0)$ can somenone give me some light on this and say who is actually correct.
Thank You.
Compute using the definition of partial derivatives:
$$ \frac{\partial f}{\partial x}(0,0) = \lim_{h \to0} \frac{f(h,0)-f(0,0)}{h} $$
Using the top line of the definition of $f$ (when $x=y$), you see that $f(0,0) = 3(0) = 0$. Using the bottom line of the definition (when $x \neq y$), you see that $f(h,0) = 0$. So the numerator in this limit is constant, and always equal to zero. So you get that $\frac{\partial f}{\partial x}(0,0) = 0$.
The reasoning for $\frac{\partial f}{\partial y}$ is essentially the same.