Question

The continuous function $f(x,y)$ is said to have saddle point at $(a,b)$ if

Where the subscripts $x,y$ etc. denote partial derivatives.

• A. $f_x(a,b)=f_y(a,b)=0;$
  $f_{xy}^2-f_{xx}{f}_{yy}<0$ at $(a,b)$
• B. $f_x(a,b)=0;f_y(a,b)=0;$
  $f_{xy}^2-f_{xx}{f}_{yy}>0$ at $(a,b)$
• C. $f_x(a,b)=0;f_y(a,b)=0;$
  $f_{xx}$ and $f_{yy}<0$ at $(a,b)$
• D. $f_x(a,b)=f_y(a,b)=0;$
  $f_{xy}^2-f_{xx}{f}_{yy}<0$ at $(a,b)$

Answer 1

The correct option is B $f_x(a,b)=0;f_y(a,b)=0;$

$f_{xy}^2-f_{xx}{f}_{yy}>0$ at $(a,b)$
$\because$ at saddle point $rt-s^2<0$