Question Bank • QB_EXP_EM_PD_035
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 $f(x, y) = x^2 + y^2 - 72x - 76y$ on $\mathbb{R}^2$ is equal to ________ (answer in integer).
Step-by-Step Solution & Explanation
Completing the square for $x$ and $y$:
$f(x, y) = (x^2 - 72x + 1296) + (y^2 - 76y + 1444) - (1296 + 1444)$
$f(x, y) = (x - 36)^2 + (y - 38)^2 - (2740)$
Since $(x - 36)^2 \ge 0$ and $(y - 38)^2 \ge 0$, the minimum occurs at $(x, y) = (36, 38)$:
$f_{\min} = -(2740) = -2740$
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