Question Bank • QB_EXP_EM_PD_026
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 $f(x, y) = x^2 + y^2 - 54x - 58y$ on $\mathbb{R}^2$ is equal to ________ (answer in integer).
Step-by-Step Solution & Explanation
Completing the square for $x$ and $y$:
$f(x, y) = (x^2 - 54x + 729) + (y^2 - 58y + 841) - (729 + 841)$
$f(x, y) = (x - 27)^2 + (y - 29)^2 - (1570)$
Since $(x - 27)^2 \ge 0$ and $(y - 29)^2 \ge 0$, the minimum occurs at $(x, y) = (27, 29)$:
$f_{\min} = -(1570) = -1570$
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