chapter 5

Question 1

When does f(x, y) have a local maximum at (x0, y0)?

Answer 1

If f(x, y) < f(x0, y0) in the vicinity of (x0, y0).

Question 2

When does f(x, y) have a local minimum at (x0, y0)?

Answer 2

If f(x, y) > f(x0, y0) in the vicinity of (x0, y0).

Question 3

What is a critical (stationary) point of f(x, y)?

Answer 3

A point (x0, y0) where: ∂f/∂x (x0, y0) = 0 and ∂f/∂y (x0, y0) = 0.

Question 4

what is Second Derivative Test for f(x, y))

Answer 4

At a critical point (x0, y0), compute: Δ = (∂²f/∂x²)(∂²f/∂y²) − (∂²f/∂x∂y)².

If Δ < 0 → saddle point

If Δ > 0 and ∂²f/∂x² < 0 → local maximum

If Δ > 0 and ∂²f/∂x² > 0 → local minimum

If Δ = 0 → test inconclusive (need higher-order Taylor expansion)

Question 5

What are the steps of the Lagrange Multiplier method?

Answer 5

Identify f(x, y, z) to maximise/minimise.

Identify constraint g(x, y, z) = 0.

Construct F(x, y, z, λ) = f(x, y, z) − λg(x, y, z).

Solve ∂F/∂x = 0, ∂F/∂y = 0, ∂F/∂z = 0, plus g(x, y, z) = 0.

Find x, y, z, λ → stationary points.

Evaluate f(x, y, z) at those points to determine maxima/minima.