Question Bank • QB_EXP_EM_PD_008
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 $f(x, y) = x^2 + y^2 - 18x - 22y$ on $\mathbb{R}^2$ is equal to ________ (answer in integer).
Step-by-Step Solution & Explanation
Completing the square for $x$ and $y$:
$f(x, y) = (x^2 - 18x + 81) + (y^2 - 22y + 121) - (81 + 121)$
$f(x, y) = (x - 9)^2 + (y - 11)^2 - (202)$
Since $(x - 9)^2 \ge 0$ and $(y - 11)^2 \ge 0$, the minimum occurs at $(x, y) = (9, 11)$:
$f_{\min} = -(202) = -202$
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