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### Section 14.5 : Lagrange Multipliers

1. Find the maximum and minimum values of $$f\left( {x,y} \right) = 10{y^2} - 4{x^2}$$ subject to the constraint $${x^4} + {y^4} = 1$$.
2. Find the maximum and minimum values of $$f\left( {x,y} \right) = 3x - 6y$$ subject to the constraint $$4{x^2} + 2{y^2} = 25$$.
3. Find the maximum and minimum values of $$f\left( {x,y} \right) = xy$$ subject to the constraint $${x^2} - y = 12$$. Assume that $$y \le 0$$ for this problem. Why is this assumption needed?
4. Find the maximum and minimum values of $$f\left( {x,y,z} \right) = {x^2} + 3{y^2}$$ subject to the constraint $${x^2} + 4{y^2} + {z^2} = 36$$.
5. Find the maximum and minimum values of $$f\left( {x,y,z} \right) = xyz$$ subject to the constraint $${x^2} + 2{y^2} + 4{z^2} = 24$$.
6. Find the maximum and minimum values of $$f\left( {x,y,z} \right) = 2x + 4y + {z^2}$$ subject to the constraints $${y^2} + {z^2} = 1$$ and $${x^2} + {z^2} = 1$$.
7. Find the maximum and minimum values of $$f\left( {x,y,z} \right) = x + y + {z^2}$$ subject to the constraints $$x + y + z = 1$$ and $${x^2} + {z^2} = 1$$.