We have the function $f(x,y)=x^2y$.
We want to find the extrema under the constraint $3x+2y=9$.
Solving for one variable at the equation of the constraint and getting then a function of one variable we get the following:
$f(x,y)$ has a local maximum at $\left (2, \frac{3}{2}\right )$ and a local minimum at $\left (0, \frac{9}{2}\right )$
When's we draw some contour lines and the constraint can we get also from there an information about the extrema?
The graph for the contour lines $f(x,y)=c$, where $c\in \{1,2,3,4,5\}$ and the constraint is the following:
Yes. Basically, when our functions and constraints are smooth, a necessary condition for a point $(x_0,y_0)$ to be an extremum over the constraint set $3x+2y=9$ is that the level set of $f(x,y)$ through $(x_0,y_0)$ touches the line $3x+2y=9$ at $(x_0,y_0)$. Equivalently (and which is easier to deal with in practice), the gradient $\nabla f(x_0,y_0)$ should be orthogonal to the line $3x+2y=9$. The intuition is that if the gradient had a component along the direction of the constraint line, we would be able to make a small move along that component and opposite to it to increase/reduce the function value.