Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$

Question

Find all local maxima and minima $f(x)=x_1 x_2 x_3 (4 - x_1 - x_2 - x_3 )$ where $x=(x_1, x_2, x_3) \in \mathbb{R}^3$

I was trying to look at Hessian matrix and use Sylwester theorem, but I see that I've $0$ in the left top corner. What is the right approach now?

Answer 1

You can use the Lagrange method such as $$\frac{\partial f}{\partial x_1}=0 \quad \frac{\partial f}{\partial x_2}=0 \quad \frac{\partial f}{\partial x_3}=0$$ and solve the system of equations.