Question Bank • QB_EXP_EM_PD_005
NAT • 2 Marks
Section 1: Engineering Mathematics
Calculus: Partial Derivatives & Maxima-Minima
Subtopic: Constrained extrema and Lagrange multipliers
Question Statement:
The minimum value of the function $f(x, y) = x^2 + y^2 - 12x - 16y$ on $\mathbb{R}^2$ is equal to ________ (answer in integer).
Step-by-Step Solution & Explanation
Completing the square for $x$ and $y$:
$f(x, y) = (x^2 - 12x + 36) + (y^2 - 16y + 64) - (36 + 64)$
$f(x, y) = (x - 6)^2 + (y - 8)^2 - (100)$
Since $(x - 6)^2 \ge 0$ and $(y - 8)^2 \ge 0$, the minimum occurs at $(x, y) = (6, 8)$:
$f_{\min} = -(100) = -100$
Practice In Authentic CBT Exam Interface
Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.
Launch CBT Mock Test