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Section 3.5

By cssp

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1. Maximum value of \dfrac{1}{2} at \displaystyle{\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)} and \displaystyle{\left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)}; minimum value of -\dfrac{1}{2} at \displaystyle{\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)} and \displaystyle{\left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)}

3. Maximum value of 10 at \left(\sqrt{2}, \sqrt{2}\right) and \left(-\sqrt{2}, -\sqrt{2}\right) ; minimum value of -2 at \left(-\sqrt{2}, \sqrt{2}\right) and \left(\sqrt{2}, -\sqrt{2}\right)

5. Local minimum of \phantom{}0 at all points of the form (0, y) , -\infty < y < \infty ; local maximum of e^{-1} at (1, 0) and (-1, 0)

7. Local maximum of 1 at (1, 1) and (-1, -1) ; saddle point at (0, 0)

9. Local minimum of \phantom{}0 at (0, 0, 0)

11. 10\hbox{ meters} \times 10\hbox{ meters} \times 10\hbox{ meters}

12. 8.43\hbox{ meters} \times 8.43\hbox{ meters} \times 8.43\hbox{ meters}

13. Maximum value of \sqrt{3} at \displaystyle{\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)} ; minimum value of -\sqrt{3} at \displaystyle{\left(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right)}

15. Minimum distance of \dfrac{2\sqrt{21}}{7} at \displaystyle{\frac{2}{7}(2, 4, 1)}

17. Hottest point: 98.28^\circ at \left(\sqrt{2}, 0, -\sqrt{2}\right) ; coldest point: 41.72^\circ at \left(-\sqrt{2}, 0, \sqrt{2}\right)

20. Local minimum of \phantom{}0 at all points of the form (x, x) , -\infty < x < \infty

24. y = 9.23x + 114.72

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