$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$. first of all I checked if (0,0) is critical point
$Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $
now my idea was to replace $y$ with $xk$ because of the reduction of $F$
,and find the hessian matrix to prove that the point is local minimum.
Is this the right way ?

A direct way to prove that $(0,0)$ is a local minimum for $F$ is just to notice that $F(0,0)=0$ and that
$$
\begin{align}
F(x,y)-F(0,0)&=2x^4-3x^2y+y^2\\\\
&=2\left( x^4-\frac32x^2y+y^2\right)\\\\
&=2\underbrace{\left( \left(x^2-\frac34y\right)^2+\frac7{16}y^2\right)}_{\large\geq \:0}
\end{align}
$$
