Question Bank • QB_EXP_EM_PD_002
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 $f(x, y) = x^2 + y^2 - 6x - 10y$ on $\mathbb{R}^2$ is equal to ________ (answer in integer).
Step-by-Step Solution & Explanation
Completing the square for $x$ and $y$:
$f(x, y) = (x^2 - 6x + 9) + (y^2 - 10y + 25) - (9 + 25)$
$f(x, y) = (x - 3)^2 + (y - 5)^2 - (34)$
Since $(x - 3)^2 \ge 0$ and $(y - 5)^2 \ge 0$, the minimum occurs at $(x, y) = (3, 5)$:
$f_{\min} = -(34) = -34$
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