Calculus III - Lagrange Multipliers (Practice Problems)

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Section 14.5 : Lagrange Multipliers

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
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
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
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
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