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Rapsinincke

Answered question

2022-07-08

Maximize

f (x, y, z) = x y + z^{2},

while

2 x - y = 0

and

x + z = 0

. Lagrange doesnt seem to work.

Answer & Explanation

poquetahr

Beginner2022-07-09Added 18 answers

Since there are two constraints, we must use

\nabla f (x, y, z) = λ \nabla g + μ \nabla h .

We have
1.

f (x, y, z) = x y + z^{2}

2.

g (x, y, z) = 2 x - y

3.

h (x, y, z) = x + z

So we get

\begin{aligned} (1) & f_{x} = λ g_{x} + μ h_{x} & ⟹ y = 2 λ + μ \\ (2) & f_{y} = λ g_{y} + μ h_{y} & ⟹ x = - λ \\ (3) & f_{z} = λ g_{z} + μ h_{z} & ⟹ 2 z = μ \end{aligned}

with your constraint equations

\begin{aligned} (4) & 2 x - y & = 0 \\ (5) & x + z & = 0 \end{aligned}

Utilizing the system of equations (1) to (5), can you take it from here?

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