Allied & ICAR Exam Bank • ALLIED_1008
MCQ • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Maxima and minima
Question Statement:
The function $f(x, y) = x^2 + 2y^2 - 4x + 4y + 7$ has a stationary point at:
Options:
-
(A) (2, -1) which is a local minimum
-
(B) (2, -1) which is a local maximum
-
(C) (-2, 1) which is a saddle point
-
(D) (2, 1) which is a local minimum
Step-by-Step Solution & Explanation
$\frac{\partial f}{\partial x} = 2x - 4 = 0 \implies x = 2$. $\frac{\partial f}{\partial y} = 4y + 4 = 0 \implies y = -1$. Second derivatives: $r = f_{xx} = 2$, $s = f_{xy} = 0$, $t = f_{yy} = 4$. $rt - s^2 = (2)(4) - 0 = 8 > 0$ and $r = 2 > 0$. Hence $(2, -1)$ is a point of local minimum.
Practice In Authentic CBT Exam Interface
Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.
Launch CBT Mock Test