Paul's Online Notes
Home / Calculus III / Applications of Partial Derivatives / Lagrange Multipliers
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### 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