Define all constant $a$ values in way that function $f(x,y)$ has critical point in $(0,0)$

Question

Problem

Define all constant $a$ values in way that function $f(x,y)$ has critical point in $(0,0)$ whe: $$ f(x,y)=(4x^2+axy+y^2)(a+x) $$

Attempt to solve

I wan to find out all constant $a$ values in a way that this function has zero gradient in point $(0,0)$ $\nabla(0,0)=0$ with what constant $a$. Gradient for this function is:

$$ \nabla f(x,y)=\begin{bmatrix} a^2y+2axy+8ax+12x^2+y^2 \\ a^2x+ax^2+2ay+2xy \end{bmatrix} $$

$$ \nabla f(0,0)=\begin{bmatrix} a^2\cdot 0 + 2\cdot 0 \cdot 0 + 8 \cdot 0 \cdot 0 + 12 \cdot 0^2 + 0^2 \\ a^2\cdot 0 + a\cdot 0 ^2 + 2 \cdot 0 \cdot 0 + 2 \cdot 0 \cdot 0 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \end{bmatrix} $$

Since value of gradient at point $(0,0)$ is not dependent on value of constant $a$. This constant can be anything when $a \in \mathbb{R}$.

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I think this is correct solution to this problem but i still have some doubt that maybe there is flaw. It would be highly appreciated if someone can tell that does this seem to be correct or not ?

Answer 1

Your answer is indeed correct.