Maxima and minima of quadratic function via the Hessian

Question

Find maxima and minima of the quadratic function $f: \Bbb R^2 \to \Bbb R$ defined by $$ f(x_1,x_2) = x_1 (x_1 - 1) x_2 $$

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I found the critical points to be $P_1(0,0)$ and $P_2(1,0)$. I know $P_2$ is a saddle point because the Hessian for this point is negative. For the other point, instead, I can't prove it is a saddle point because both the Hessian and the Quadratic form are equal to zero, and I can't seem to find a theorem on my book or on the internet that takes into account this case. I would like a hint on how to solve this, thanks in advance.
Hessian: $$detHf=-4x_1$$ Quadratic form of a critical point $x^0$: $$Q(h_1,h_2)=(1/2)[2x^0_2h_1^2+4x^0_1h_1h_2]$$

Answer 1

For $0 < h <1$ you have

$$\begin{aligned} f(h,h) = h^2(h-1) <0\\ f(h,-h)= -h^2(h-1) >0 \end{aligned}$$

Proving that $(0,0)$ which is neither a minimum nor a maximum is a saddle point.