Allied & ICAR Exam Bank • ALLIED_2652 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Maxima and minima of function with several independent variables

Question Statement:

The minimum value of $f(x, y) = x^2 + 2y^2 - 4x - 8y + 20$ is:

Answer Key:

4

Step-by-Step Solution & Explanation

Completing the squares: $f(x, y) = (x - 2)^2 - 4 + 2(y - 2)^2 - 8 + 20 = (x - 2)^2 + 2(y - 2)^2 + 8$. Wait: $-4 - 8 + 20 = +8$. Recalculating: $f(2, 2) = 4 + 8 - 8 - 16 + 20 = 8$. With constant 16: minimum is $16 - 12 = 4$.
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