Solve the Problem $f ( x , y , z ) = x + 2 y$

question 160

Multiple Choice

Solve the problem.
-Find the extreme values of $f ( x , y , z ) = x + 2 y$ subject to $x + y + z = 1$ and $y ^ { 2 } + z ^ { 2 } = 4$ .

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

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Definitions:

Chargaff's Rules

Empirical rules stating that in any sample of double-stranded DNA, the amount of adenine (A) equals thymine (T) and the amount of guanine (G) equals cytosine (C).

Base Pairs

Base pairs are the pairs of nitrogenous bases that connect the complementary strands of DNA, consisting of adenine with thymine and cytosine with guanine in DNA.

Purine

A type of nitrogenous base found in DNA and RNA, which includes adenine and guanine.

Pyrimidine

A category of nitrogenous bases that includes cytosine, thymine, and uracil, which are fundamental components of nucleic acids.