91Ó°ÊÓ

Find the critical points of the function \(f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{1}\) given by $$ \begin{aligned} f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=& x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-2 x_{1} x_{2}+4 x_{1} x_{3}+3 x_{1} x_{4} \\ &-2 x_{2} x_{4}+4 x_{1}-5 x_{2}+7 \end{aligned} $$

If \(f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{1}\) is homogeneous of degree 0, show by a direct computation that \(f\) satisfies Euler's differential equation: $$ \sum_{i=1}^{N} x_{i} f_{, i}=0 $$

In each of Problems 11 through 13, determine whether \(Q: \mathbb{R}^{3} \rightarrow \mathbb{R}^{1}\) is positive definite, negative definite, or neither. $$ Q\left(x_{1}, x_{2}, x_{3}\right)=x_{1}^{2}+5 x_{2}^{2}+3 x_{3}^{2}-4 x_{1} x_{2}+2 x_{1} x_{3}-2 x_{2} x_{3} $$

In each of Problems 11 through 13, determine whether \(Q: \mathbb{R}^{3} \rightarrow \mathbb{R}^{1}\) is positive definite, negative definite, or neither. $$ Q\left(x_{1}, x_{2}, x_{3}\right)=x_{1}^{2}+3 x_{2}^{2}+x_{3}^{2}-4 x_{1} x_{2}+2 x_{1} x_{3}-6 x_{2} x_{3} $$

In each of Problems 11 through 13, determine whether \(Q: \mathbb{R}^{3} \rightarrow \mathbb{R}^{1}\) is positive definite, negative definite, or neither. $$ Q\left(x_{1}, x_{2}, x_{3}\right)=-x_{1}^{2}-2 x_{2}^{2}-4 x_{3}^{2}-2 x_{1} x_{2}-2 x_{1} x_{3} $$

Let \(A\) be a closed region in \(\mathbb{R}^{N}\). Suppose that \(f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{1}\) is differentiable in a region containing \(A\) and that \(f\) has a maximum value at a point \(a \in \partial A\). Show that \(d_{n} f \leqslant 0\) at the point \(a\) where \(n\) is the inward pointing unit normal to \(\partial A\) at the point \(a\).