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Section Solutions

Critical Points

    1. \((0,0,1)\) is a max

    2. none, why?

    3. \((0,1, 4)\) is a min

    4. \((0,0,0)\) is a min

    1. \((\pm 1,0,2)\) are local minima and \((0,\pm 1, -1)\) are local maxima

    2. all points to consider should be: \((2,-1/3)\text{,}\) \((0,0\)), \((3,0)\text{,}\) \((3,-3)\text{,}\) \((0,-3)\text{,}\) \((2,0)\text{,}\) \((0,-1/3)\text{,}\) \((3,-1/3)\text{,}\) and \((2,-3)\text{;}\) \((2,-1/3,-22/3)\) is a min; \((0,-3,18)\) is a max

  3. \((-8/3, 5/2)\) and \(\sqrt{481}/6\)

  4. \((3,-320)\)

    1. \((3,\pm6,-54)\) are saddles and \((0,0,0)\) are all maxima

    2. \((0,1,9/2)\) (D is long — use software!)

    3. icky algebra — use software!

Optimization and Lagrange Multipliers

  1. \((1/2,1/2,1/2)\) is the max, each of \((1,0,0)\) and \((0,1,0)\) is a min

  2. \((0,0,1)\) is the max, each of \((1,0,0)\) and \((0,1,0)\) is a min

  3. there is one saddle on the interior, but no extrema on the interior; the extrema occur on the boundaries at \((0,0),\) \((\pi/2,5),\) \((2,0),\) and \((\pi/2,0)\)

  4. \((12,12,288)\)

  5. \((0, \pm 2)\) and \(\dsp (\pm\frac{\sqrt{15}}{2},1/2)\)

  6. \(x=41, y = 41, z= 41\)

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