First slide

Theory of expressions

Question

If $x, y, z \in R$ then the least value of the
expression $E = 3 x^{2} + 5 y^{2} + 4 z^{2} - 6 x + 20 y - 8 z - 3$ is:

Moderate

A

-15
B

-30
C

-45
D

-60

Solution

We can write
$\begin{array}{l} E = 3 (x^{2} - 2 x + 1) + 5 (y^{2} + 4 y + 4) \\ + 4 (z^{2} - 2 z + 1) - 30 \\ = 3 (x - 1)^{2} + 5 (y + 2)^{2} + 4 (z - 1)^{2} - 30 \geq - 30 \end{array}$
The least value is attained when
$x = 1, y = - 2, z = 1$

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