<s>[INST] I need help finding the values of the constants of a function $f\left(x,y\right)=ax^2y+bxy+2xy^2+c$
I need to find the values of the constants a, b, c so that the function has a point $P\left(\frac{2}{3},\frac{1}{3}\right)$ and and minimum value $-\frac{1}{9}$
My solution
$f´(x)=2axy+by+2y^2$
$f´(y)=2x^2+bx+4xy$
Evaluate $P\left(\frac{2}{3},\frac{1}{3}\right)$ in $f´(x)=2axy+by+2y^2$
$2a\frac{2}{3}\cdot \frac{1}{3}+b\cdot \frac{1}{3}+2\left(\frac{1}{3}\right)^2=0$
$b=\frac{-4a}{3}-\frac{2}{2}$
Evaluate  $P\left(\frac{2}{3},\frac{1}{3}\right)$ and $b=\frac{-4a}{3}-\frac{2}{2}$ in $f´(y)=2x^2+bx+4xy$
$a\left(\frac{2}{3}\right)^2+\frac{-4a}{3}-\frac{2}{3}\cdot \frac{2}{3}+4\cdot \frac{2}{3}\cdot \frac{1}{3}$
$a=\frac{1}{2}$
Evaluate $a=\frac{1}{2}$ in $b=\frac{-4a}{3}-\frac{2}{2}$
$b=\frac{-4}{3}$
The main problem is the constant c because I can't find a way to find that value and I need to know if what I did is right
 [/INST] Plug in your values of $a$ and $b$ into $f$ to get your function of $x$ and $y$. Then you want the value of the function to be $-1/9$ at the point $P$, so solve the equation $f(P) = -1/9$ for $c$.
</s>