First slide

Theory of expressions

Question

$The least value of the expression x^{2} + 4 y^{2} + 3 z^{2} - 2 x - 12 y - 6 z + 14 is$

Moderate

A

1
B

No least value
C

0
D

None of these

Solution

$Let f (x, y, z) = x^{2} + 4 y^{2} + 3 z^{2} - 2 x - 12 y - 6 z + 14$
$x^{2} - 2 x + 1 - 1 + 4 y^{2} - 12 y + 9 - 9 + 3 z^{2} - 6 z + 3 - 3 + 14$
$= (x - 1)^{2} + (2 y - 3)^{2} + 3 (z - 1)^{2} + 1$
$For least value of f (x, y, z)$
$x - 1 = 0; 2 y - 3 = 0 and z - 1 = 0$
$\begin{array}{l} ∴ x = 1; y = 3 / 2; z = 1 \\ Hence least value of f (x, y, z) is f (1, 3 / 2, 1) = 1 \end{array}$

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