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**1. Homework Statement**

The task is to find the extreme values (and their nature) of the polynomial function . $$f(\vec{x})=x_1x_2+x_1^2+x_2^2+x_3^3+x_4^4.$$

**3. The Attempt at a Solution**

The critical point is ##a=(0,0,0,0)##, which is the solution to ##\nabla{f(a)}=0.## If we form the Hessian matrix $$H_f=\begin{bmatrix}

2& 1 & 0 & 0 \\

1& 2 & 0 & 0 \\

0 & 0 & 0 &0 \\

0 & 0 & 0 & 0 \\

\end{bmatrix}$$ ,it's easy to see that ##det(H_f)=0##, thus the product of the eigenvalues is zero. This test is inconclusive.

How would one determine the extreme values of ##f## in this case? What is the general approach?

I have also tried

## f(\vec{h})=(h_1+\frac{h_2}{2})^2+(h_4^2)^2+h_3^3,##where the term ##h_3^3##

is giving trouble. I cannot factor ##f## such that ##f(\vec{h})=(\alpha_1(\vec{h}))^2+...+(\alpha_k(\vec{h}))^2-(\alpha_{k+1}(\vec{h}))^2-...-(\alpha_{k+l}(\vec{h}))^2,## where ##\alpha_i## are linearly independent linear functions.

Graphical interpretation is not an option either, since we are in ##\mathbb{R^4}.##In addition I have tried different values of ##x_1,...,x_4## to try to determine if ##a## is min/max point at all without success.

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