\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({x}-{y}\right)}{\left({16}-{x}{y}\right)}\)

\(\displaystyle{f}_{{x}}={\left({16}-{x}{y}\right)}+{\left({x}-{y}\right)}{\left(-{y}\right)}={16}-{x}{y}+{y}^{{2}}-{x}{y}\)

\(\displaystyle{f}_{{y}}={\left({x}{y}-{16}\right)}+{\left({x}-{y}\right)}{\left(-{x}\right)}={x}{y}-{16}-{x}^{{2}}+{x}{y}\)

\(\displaystyle{f}_{{x}}={0}\ \Rightarrow\ {y}^{{2}}-{2}{x}{y}-{16}={0};\ \Rightarrow{y}^{{2}}={x}^{{2}}\)

\(\displaystyle{f}_{{y}}={0}\ \Rightarrow\ {x}^{{2}}-{2}{x}{y}-{16}={0};\ \Rightarrow{y}=\pm{x}\)

\(\displaystyle{y}={x}\ \Rightarrow\ {x}^{{2}}-{2}{x}^{{2}}+{16}={0}\ \Rightarrow\ {x}^{{2}}={16}\ \Rightarrow\ {x}=\pm{4}\)

\(\displaystyle{y}=-{x}\ \Rightarrow\ {x}^{{2}}+{2}{x}^{{2}}+{16}={0}\ \Rightarrow\ {x}^{{2}}=-{\frac{{{16}}}{{{3}}}}\) - not possible

\(\displaystyle{\left({x},{y}\right)}={\left(-{4},-{4}\right)},{\left({4},{4}\right)}\)

\(\displaystyle{f}_{{\times}}=-{2}{y};\ {f}_{{{y}{y}}}={2}{x};\ {f}_{{{x}{y}}}=-{2}{x}+{2}{y}\)

\(\displaystyle{D}=-{4}{x}{y}-{4}{\left({x}-{y}\right)}^{{2}}\)

\(\displaystyle{D}{\left({4},-{4}\right)}=-{4}{\left({16}\right)}=-{64}{ < }{0}\)

\(\displaystyle{D}{\left({4},{4}\right)}=-{4}{\left({16}\right)}=--{64}{ < }{0}\)

\(\displaystyle{f}_{{x}}={\left({16}-{x}{y}\right)}+{\left({x}-{y}\right)}{\left(-{y}\right)}={16}-{x}{y}+{y}^{{2}}-{x}{y}\)

\(\displaystyle{f}_{{y}}={\left({x}{y}-{16}\right)}+{\left({x}-{y}\right)}{\left(-{x}\right)}={x}{y}-{16}-{x}^{{2}}+{x}{y}\)

\(\displaystyle{f}_{{x}}={0}\ \Rightarrow\ {y}^{{2}}-{2}{x}{y}-{16}={0};\ \Rightarrow{y}^{{2}}={x}^{{2}}\)

\(\displaystyle{f}_{{y}}={0}\ \Rightarrow\ {x}^{{2}}-{2}{x}{y}-{16}={0};\ \Rightarrow{y}=\pm{x}\)

\(\displaystyle{y}={x}\ \Rightarrow\ {x}^{{2}}-{2}{x}^{{2}}+{16}={0}\ \Rightarrow\ {x}^{{2}}={16}\ \Rightarrow\ {x}=\pm{4}\)

\(\displaystyle{y}=-{x}\ \Rightarrow\ {x}^{{2}}+{2}{x}^{{2}}+{16}={0}\ \Rightarrow\ {x}^{{2}}=-{\frac{{{16}}}{{{3}}}}\) - not possible

\(\displaystyle{\left({x},{y}\right)}={\left(-{4},-{4}\right)},{\left({4},{4}\right)}\)

\(\displaystyle{f}_{{\times}}=-{2}{y};\ {f}_{{{y}{y}}}={2}{x};\ {f}_{{{x}{y}}}=-{2}{x}+{2}{y}\)

\(\displaystyle{D}=-{4}{x}{y}-{4}{\left({x}-{y}\right)}^{{2}}\)

\(\displaystyle{D}{\left({4},-{4}\right)}=-{4}{\left({16}\right)}=-{64}{ < }{0}\)

\(\displaystyle{D}{\left({4},{4}\right)}=-{4}{\left({16}\right)}=--{64}{ < }{0}\)