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Problem 152

Find the critical points and the nature of each critical point (i.e., relative maximum, relative minimum, or saddle point) for: a) \(f(x, y)=x^{2}-2 x y+2 y^{2}+x-5\) b) $\mathrm{f}(\mathrm{x}, \mathrm{y})=(1-\mathrm{x})(1-\mathrm{y})(\mathrm{x}+\mathrm{y}-1)$.

Expert verified

The critical points and their nature for the given functions are:
a) For \(f(x, y)=x^{2}-2 x y+2 y^{2}+x-5\), there is one critical point at \(\left(-\frac{1}{2}, 0\right)\), which is a relative minimum.
b) For \(f(x, y)=(1-x)(1-y)(x+y-1)\), there are two critical points at \((0,0)\) and \((1,1)\). However, the second partial derivatives test is inconclusive, and we cannot determine their nature using this method.

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Chapter 6

Minimize \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) subject to $\mathrm{g}(\mathrm{x}, \mathrm{y})=(\mathrm{x}-1)^{3}-\mathrm{y}^{2}=0$ a) graphically b) using the Lagrangian multiplier method.

Chapter 6

Let \(\mathrm{p}, \mathrm{q}\) be positive real numbers such that \(\mathrm{p}^{-1}+\mathrm{q}^{-1}+=1\). Consider the function $\mathrm{f}=\mathrm{a}^{\rightarrow} \cdot \mathrm{X}^{-}={ }^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{a}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}$, for any \(\mathrm{a}^{\rightarrow}\), with all \(\mathrm{a}_{\mathrm{i}}>0\), on the compact set \(\mathrm{S}=\left\\{\mathrm{X}^{-} \mid\right.\) all $\left.\mathrm{x}_{\mathrm{i}} \geq 0, \mathrm{x}^{\mathrm{p}}_{\mathrm{i}}+\ldots+\mathrm{x}^{\mathrm{p}}_{\mathrm{n}}=1\right\\}$. Show that the maximum value of \(\mathrm{a}^{\rightarrow} \cdot \mathrm{X}^{-}\) occurs at a point where all \(\mathrm{x}_{\mathrm{i}}>0\). Then, using this method, derive Holder's inequality.

Chapter 6

Find a relative minimum for \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\) $$ 2 x^{2}+2 y^{2}+2 z^{2}-2 x z-2 y z-6 x+2 y+8 z+14 $$

Chapter 6

Let the function \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) be continuous and have continuous first and second partial derivatives in a region \(R .\) Let \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\) be an interior point of \(\mathrm{R}\) for which ( $\left.\partial \mathrm{f} / \partial \mathrm{x}\right)=0$ \((\partial \mathrm{f} / \partial \mathrm{y})=0 .\) Given the condition $\left[\mathrm{f}_{12}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\right]^{2}-\mathrm{f}_{11}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)$ \(\mathrm{f}_{22}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)<0\) and \(\mathrm{f}_{11}\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)<0\) prove that \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) has a relative maximum at \(\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)\).

Chapter 6

\(\mathrm{M}\) where \(\mathrm{p}_{1}=1, \mathrm{p}_{2}=2\) and \(\mathrm{M}=10 .\) Check the second-order conditions to verify that the solution is indeed a maximum.

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