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Minimize the function

$$ Z = \frac{1}{2}x_1² + x_2² +x_3²$$ subject to

$$x_1 - x_2 = 0$$ and $$x_1 + x_2 +x_3 =1$$

Steven01123581321
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Cookey Cookey
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Aug 10 '21 at 12:27
  • Lagrange multipliers would work – FShrike Aug 10 '21 at 14:03

1 Answers1

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Using equations $2$ and $3,$ solve for $x_1,x_2$ in terms of $x_3$. $$x_1=x_2=\dfrac{1-x_3}{2}$$Substituting in $1^\text{st}$ equation, we get a quadratic equation in terms of $x_3$.$$ Z=\dfrac{1}{4}(7x_3^2-6x_3+3)$$ Hence the minimum value exists at $x=-\dfrac{b}{2a}=\dfrac{3}{7}$ and the minimum value is $-\dfrac{D}{4a}=\dfrac{3}{7}$ or simply solve for $f'(x)=0$ and substitute the value of $x$ in $Z$ as $f''(x)=\dfrac{7}{2}$ which is clearly positive so we would obtain a minima (required).

UNAN
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