Locate and classify all the critical points of the function:
h(x, y) = 9x^2y − 2x^2 − 4y^2

Question

Locate and classify all the critical points of the function:
h(x, y) = 9x^2y − 2x^2 − 4y^2

anonymous · Answer

The gradient of h is
$$\left\{18 x y-4 x,9 x^2-8 y\right\}
$$
The gradient is zero at
$$

p1=( 0,  0) \
p2=\left( -\frac{4}{9} , \frac{2}{9} \right)\
p3=\left(  \frac{4}{9} , \frac{2}{9} \right)\


$$

Compute the Hessian Determinant at each point and try to classify these points

anonymous · Answer

and also the second derivative test'

anonymous · Answer

The hessian Determinant at (ab) is defined as
$$h_{\text{xx}}(a,b)
   h_{\text{yy}}(a,b)-h_{\text{xy}}^2(a,b)
$$

anonymous · Answer

In your particular function h, The hessian Determinant at (a,b) is  $$
-324 a^2-144 b+32
$$

anonymous · Answer

Look how your graph look near (0,0)

anonymous · Answer

Near the point 

$$ 
\left ( \frac{4}{9}, \frac{2}{9} \right)
$$
The graph looks like the attached graph.