We have, \(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{{x}{y}}}{{7}}\)

To find the critical point:

Find the partial derivative and equate to zero:

Now,

Differntiate f(x,y) with respect to x and equate to zero

\(\displaystyle{{f}_{{x}}{\left({x},{y}\right)}}=\frac{{y}}{{7}}={0}\)

\(\displaystyle\Rightarrow{y}={0}\)

Differntiate f(x,y) with respect to y and equate to zero

\(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}=\frac{{x}}{{7}}={0}\)

\(\displaystyle\Rightarrow{x}={0}\)

Hence, the critical point is (0,0)

Now we have to find the relative extrema.

Use:

\(\displaystyle{D}{\left({x},{y}\right)}={{f}_{{\times}}{\left({x},{y}\right)}}{{f}_{{{y}{y}}}{\left({x},{y}\right)}}−{\left({{f}_{{{x}{y}}}{\left({x},{y}\right)}}\right)}^{{2}}\)

Find D(0,0):

Now

\(\displaystyle{{f}_{{\times}}{\left({0},{0}\right)}}={0}\)

\(\displaystyle{{f}_{{{y}{y}}}{\left({0},{0}\right)}}={0}\)

\(\displaystyle{{f}_{{{x}{y}}}{\left({0},{0}\right)}}=\frac{{1}}{{7}}\)

Therefore,

\(\displaystyle{D}{\left({0},{0}\right)}={0}\times{0}-{\left(\frac{{1}}{{7}}\right)}^{{2}}=\frac{{1}}{{49}}{<}{0}\)</span>

Hence, by the \(\displaystyle{2}^{{{n}{d}}}\) derivative test:

f(x,y) neither max. nor min at (0,0) which means (0,0) is saddle point.

To find the critical point:

Find the partial derivative and equate to zero:

Now,

Differntiate f(x,y) with respect to x and equate to zero

\(\displaystyle{{f}_{{x}}{\left({x},{y}\right)}}=\frac{{y}}{{7}}={0}\)

\(\displaystyle\Rightarrow{y}={0}\)

Differntiate f(x,y) with respect to y and equate to zero

\(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}=\frac{{x}}{{7}}={0}\)

\(\displaystyle\Rightarrow{x}={0}\)

Hence, the critical point is (0,0)

Now we have to find the relative extrema.

Use:

\(\displaystyle{D}{\left({x},{y}\right)}={{f}_{{\times}}{\left({x},{y}\right)}}{{f}_{{{y}{y}}}{\left({x},{y}\right)}}−{\left({{f}_{{{x}{y}}}{\left({x},{y}\right)}}\right)}^{{2}}\)

Find D(0,0):

Now

\(\displaystyle{{f}_{{\times}}{\left({0},{0}\right)}}={0}\)

\(\displaystyle{{f}_{{{y}{y}}}{\left({0},{0}\right)}}={0}\)

\(\displaystyle{{f}_{{{x}{y}}}{\left({0},{0}\right)}}=\frac{{1}}{{7}}\)

Therefore,

\(\displaystyle{D}{\left({0},{0}\right)}={0}\times{0}-{\left(\frac{{1}}{{7}}\right)}^{{2}}=\frac{{1}}{{49}}{<}{0}\)</span>

Hence, by the \(\displaystyle{2}^{{{n}{d}}}\) derivative test:

f(x,y) neither max. nor min at (0,0) which means (0,0) is saddle point.