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Ch 8.5 - 2E
Chapter 8, Problem 2E(choose chapter or problem)
Given points \(\mathbf{p}_{1}=\left[\begin{array}{r}0 \\ -1\end{array}\right], \mathbf{p}_{2}=\left[\begin{array}{l}2 \\ 1\end{array}\right] \text {, and } \mathbf{p}_{3}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\) in \(\mathbb{R}^{2}\), let \(S=\operatorname{conv}\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}\). For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f (x) = m.
a. \(f\left(x_{1}, x_{2}\right)=x_{1}+x_{2}\)
b. \(f\left(x_{1}, x_{2}\right)=x_{1}-x_{2}\)
c. \(f\left(x_{1}, x_{2}\right)=-2 x_{1}+x_{2}\)
Questions & Answers
QUESTION:
Given points \(\mathbf{p}_{1}=\left[\begin{array}{r}0 \\ -1\end{array}\right], \mathbf{p}_{2}=\left[\begin{array}{l}2 \\ 1\end{array}\right] \text {, and } \mathbf{p}_{3}=\left[\begin{array}{l}1 \\ 2\end{array}\right]\) in \(\mathbb{R}^{2}\), let \(S=\operatorname{conv}\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}\). For each linear functional f, find the maximum value m of f on the set S, and find all points x in S at which f (x) = m.
a. \(f\left(x_{1}, x_{2}\right)=x_{1}+x_{2}\)
b. \(f\left(x_{1}, x_{2}\right)=x_{1}-x_{2}\)
c. \(f\left(x_{1}, x_{2}\right)=-2 x_{1}+x_{2}\)
ANSWER:Solution 2E1. And note that are the points in S at w