I want to find the equation of the tangent of the curve $f(x,y)=x^5+x^4+y^2$ at $(0,0)$.
I have tried the following:
$\nabla f (x,y)=(5x^4+4x^3, 2y)$
$\nabla f(0,0)=(0,0)$
The equation of the tangent is given by the relation: $\nabla f(0,0) \cdot (x-0, y-0)=0 \Rightarrow (0,0) \cdot (x,y)=0 \Rightarrow 0 \cdot x + 0 \cdot y=0$
Could you tell me if it is right? If so, which the equation of the tangent?
Edit:
There is an example with tangent in my notes: The tangents of $f(x,y)=-4x^2y^2+(x^2+y^2)^3$ at $(0,0)$ are $x=0$ and $y=0$. The tangents of $g(x,y)=-y^3+3x^2y+(x^2+y^2)^2$ at $(0,0)$ are $y=0, \sqrt{3} x+y=0, \sqrt{3} x-y=0$.
How can I find it in this case?