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Section 3-5 : Lagrange Multipliers

  1. Find the maximum and minimum values of \(f\left( {x,y} \right) = 81{x^2} + {y^2}\) subject to the constraint \(4{x^2} + {y^2} = 9\). Solution
  2. Find the maximum and minimum values of \(f\left( {x,y} \right) = 8{x^2} - 2y\) subject to the constraint \({x^2} + {y^2} = 1\). Solution
  3. Find the maximum and minimum values of \(f\left( {x,y,z} \right) = {y^2} - 10z\) subject to the constraint \({x^2} + {y^2} + {z^2} = 36\). Solution
  4. Find the maximum and minimum values of \(f\left( {x,y,z} \right) = xyz\) subject to the constraint \(x + 9{y^2} + {z^2} = 4\). Assume that \(x \ge 0\) for this problem. Why is this assumption needed? Solution
  5. Find the maximum and minimum values of \(f\left( {x,y,z} \right) = 3{x^2} + y\) subject to the constraints \(4x - 3y = 9\) and \({x^2} + {z^2} = 9\). Solution