Solve the Problem $f ( x , y , z ) = x y z$

question 257

Multiple Choice

Solve the problem.
-Find the extreme values of $f ( x , y , z ) = x y z$ subject to $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and $x + y = 2$ .

• A) Maximum: \(\sqrt { 2 }\) at \(( 1 , - 1 , - \sqrt { 2 } )\); minimum: \(- \sqrt { 2 }\) at \(( 1 , - 1 , \sqrt { 2 }\),
• B) Maximum: \(\sqrt { 3 }\) at \(( 1,1 , \sqrt { 3 } )\); minimum: \(- \sqrt { 3 }\) at \(( 1,1 , - \sqrt { 3 } )\)
• C) Maximum: \(\sqrt { 3 }\) at \(( 1 , - 1 , - \sqrt { 3 } )\); minimum: \(- \sqrt { 3 }\) at \(( 1 , - 1 , \sqrt { 3 }\),
• D) Maximum: \(\sqrt { 2 }\) at \(( 1,1 , \sqrt { 2 } )\); minimum: \(- \sqrt { 2 }\) at \(( 1,1 , - \sqrt { 2 } )\)

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Definitions:

Intellectual

Relating to the intellect or involving the ability to think and understand at a high level of complexity.

Reciprocal

Characterized by mutual exchange or influence, in which actions or emotions are mirrored or returned by others.

Self-locomotion

The capacity of an individual to move independently, often referring to the developmental stage in infants.

Babbling

The stage in child language development involving the production of repetitive syllables and sounds, a key precursor to language acquisition.