$${f_x}\left( {a,\,b} \right) = {f_y}\left( {a,\,b} \right) = 0$$
$$f_{xy}^2 - {f_{xx}}{f_{yy}} < 0$$ at $$(a, b)$$

$${f_x}\left( {a,\,b} \right) = 0,{f_y}\left( {a,\,b} \right) = 0$$
$$f_{xy}^2 - {f_{xx}}{f_{yy}} > 0$$ at $$(a, b)$$

$${f_x}\left( {a,\,b} \right) = 0,{f_y}\left( {a,\,b} \right) = 0,$$
$${f_{xx}}$$ and $${f_{yy}} < 0$$ at $$(a, b)$$

$${f_x}\left( {a,\,b} \right) = 0,{f_y}\left( {a,\,b} \right) = 0$$
$$f_{xy}^2 - {f_{xx}}{f_{yy}} = 0\,\,$$ at $$(a, b)$$

$$1 - {{{x^2}} \over {2!}} + {{{x^4}} \over {4!}} - ......$$

$$1 + {{{x^2}} \over {4!}} + {{{x^4}} \over {4!}} + ......$$

$$x + {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} + ......$$

$$x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - ......$$

Taylor's series and Fourier's series

Taylor's series and not by Fourier's series

Neither Taylor's series nor Fourier's series

not by Taylor's series, but by Fourier's series