Lagrange Multipliers
Optimise with constraints.
Key idea
Optimise with constraints.
In depth
To optimise f subject to g = c, solve ∇f = λ∇g and g = c. At an extremum, the level curve of f is tangent to the constraint, so the gradients are parallel.
Worked example
Maximise xy subject to x + y = 10: ⟨y, x⟩ = λ⟨1, 1⟩ → x = y = 5, product 25.
Formula
Real-world uses
- Maximising the area of a box with a fixed material budget
- Economics with budget constraints
Common mistake
Forgetting to use the constraint equation.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lagrange Multipliers?
Optimise with constraints. It is a topic at the expert level in Calculus.
What is the formula for Lagrange Multipliers?
The key formula is ∇f = λ∇g.
Where is Lagrange Multipliers used in real life?
Real-world examples: Maximising the area of a box with a fixed material budget; Economics with budget constraints.
What should I learn before Lagrange Multipliers?
Make sure you are comfortable with Tangent Planes, the previous topic in Calculus.
What is a common mistake with Lagrange Multipliers?
Forgetting to use the constraint equation.
How can I practise Lagrange Multipliers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lagrange Multipliers?
Move on to Multivariable Extrema next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.