Allied & ICAR Exam Bank • ALLIED_2655MSQ • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Maxima and minima of function with several independent variables
Question Statement:
Which of the following conditions guarantee that a critical point $(x_0, y_0)$ of $f(x, y)$ is a saddle point?
Options:
(A) The Hessian determinant $\det(H) < 0$
(B) The eigenvalues of the Hessian matrix have opposite signs
(C) $\det(H) > 0$ and $f_{xx} = 0$
(D) $f_{xx} f_{yy} - (f_{xy})^2 < 0$
Answer Key:
A, B, D
Step-by-Step Solution & Explanation
Both $\det(H) < 0$ and eigenvalues with opposite signs define a saddle point in 2D. (C) is impossible because if $\det(H) = f_{xx}f_{yy} - f_{xy}^2 > 0$ and $f_{xx}=0$, then $-f_{xy}^2 > 0$ which has no real solution.