Multivariable Optimization & Lagrange Multipliers
Key ideas for locating and classifying critical points of multivariable functions and handling constraints.
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Questions Covered in This Set
10 cards to master
What condition must hold at an interior local extremum of a differentiable f: ℝⁿ→ℝ?
∇f(a) = 0 — the gradient vanishes, so the tangent plane to z=f(x,y) is horizontal. Proof idea: restrict to lines g(t)=f(a+tu); g'(0)=∇f(a)·u=0 for all u.
What is a saddle point, and why has it no 1-D analogue?
A critical point where f increases along some directions and decreases along others (e.g. f=x²−y² at the origin). In 1-D the second derivative only has one sign to check; in higher dimensions directions can disagree.
Why does the Hessian completely determine local shape at a critical point?
The second-order Taylor expansion is f(a+h) ≈ f(a) + ½ hᵀHh; the linear term vanishes since ∇f(a)=0, so the quadratic form hᵀHh governs behavior.
Why is the Hessian symmetric, and what does that buy you?
Clairaut's theorem: mixed partials commute for C² functions. Symmetry gives real eigenvalues and an orthonormal eigenbasis, so the form becomes λ₁h₁²+⋯+λₙhₙ².
Classify a critical point by the eigenvalue signs of H.
All λ>0 (positive definite) → local min; all λ<0 (negative definite) → local max; mixed signs (indefinite) → saddle; some λ=0 → inconclusive, higher-order terms decide.
State the 2-variable discriminant test.
D = det H = f_xx f_yy − f_xy². D>0 & f_xx>0 → local min; D>0 & f_xx<0 → local max; D<0 → saddle; D=0 → no information.
Analyze f(x,y)=x³+y³−3xy.
∇f=(3x²−3y, 3y²−3x) → critical points (0,0) and (1,1). H=[[6x,−3],[−3,6y]], D=36xy−9. At (0,0): D=−9 → saddle. At (1,1): D=27>0, f_xx=6>0 → local min with f=−1.
Does a local minimum guarantee a global minimum?
No. For x³+y³−3xy, letting x→−∞ with y=0 gives f→−∞. On unbounded domains extrema need not exist.
What is the geometric condition at a constrained extremum of f subject to g=c?
You can't improve f by moving along the constraint: ∇f·v=0 for every tangent direction v of the level set, so ∇f is normal to the constraint — parallel to ∇g.
State the Lagrange multiplier equations (codimension one).
∇f = λ∇g together with g(x)=c; solve this system for x and λ, then compare f-values at the solutions.
