Questions: Critical Points and Classification of Extrema

Question 1

At a critical point of f(x, y), the second partial derivatives are f_xx = 3, f_yy = 1, and f_xy = 2. How should this point be classified?

• A. Local minimum, because f_xx > 0 and f_yy > 0
• B. Local maximum, because the mixed partial f_xy is positive
• C. Saddle point, because D = f_xx · f_yy − (f_xy)² = 3 − 4 = −1 < 0
• D. Inconclusive, because f_xx and f_yy have different magnitudes

Answer: (C)

The discriminant D = f_xx · f_yy − (f_xy)² = (3)(1) − (2)² = 3 − 4 = −1 < 0, which classifies this as a saddle point regardless of the signs of f_xx and f_yy. Option A is the most tempting error: students see positive f_xx and f_yy and conclude 'upward curvature everywhere.' But the mixed partial term in D captures the twisting of the surface that produces a saddle. D < 0 means the principal curvatures have opposite signs — the surface curves up in some directions and down in others.

Question 2

You are maximizing f(x, y) over a closed bounded region. You find all interior critical points where ∇f = 0 and apply the second derivative test. What must you still do before identifying the absolute maximum?

• A. Nothing — the largest value at any local maximum candidate is the absolute maximum
• B. Verify that D > 0 at each critical point to confirm they are true extrema
• C. Evaluate f on the boundary of the region and compare all values
• D. Check whether the function is concave down globally by verifying f_xx < 0 everywhere

Answer: (C)

By the extreme value theorem, a continuous function on a closed bounded region attains its absolute maximum and minimum — but the maximum might occur on the boundary, not at an interior critical point. You must parameterize the boundary (it's typically a curve), optimize f along it using single-variable methods, and then compare those boundary values against the interior critical point values. Stopping at interior critical points misses any extrema that occur on the edge of the domain.

Question 3

A saddle point of f(x, y) is a critical point where the gradient ∇f is zero but the point is neither a local maximum nor a local minimum.

• T. True
• F. False

Answer: True

At a saddle point, ∇f = 0 exactly as at a local extremum — the gradient condition alone cannot distinguish the cases. A saddle point looks like a mountain pass: the function rises in some directions from the point and falls in others. The discriminant D = f_xx · f_yy − (f_xy)² < 0 detects this by signaling that the surface has opposite curvatures in different directions.

Question 4

If the discriminant D > 0 at a critical point, then that point is a local minimum.

• T. True
• F. False

Answer: False

D > 0 means the curvature has a consistent sign in all directions (a pure bowl shape), but it does not specify which way the bowl opens. If D > 0 and f_xx > 0, the bowl opens upward — local minimum. If D > 0 and f_xx < 0, the bowl opens downward — local maximum. Both conditions together are needed: D > 0 rules out saddle points, and the sign of f_xx reveals whether the bowl is a minimum or maximum.

Question 5

Why is the condition ∇f = 0 necessary but not sufficient to conclude that a point is a local minimum of f(x, y)?

Model answer: A local minimum requires the function to increase in every direction away from the point, which forces both partial derivatives to be zero — hence ∇f = 0 is necessary. But saddle points also have ∇f = 0: the function rises in some directions and falls in others, so there is no local extremum despite the zero gradient. The gradient test cannot distinguish between local minima, local maxima, and saddle points. Additional information about the second-order behavior — specifically the discriminant D and the sign of f_xx — is required to classify which type of critical point it is.

This mirrors the one-variable situation: f′(x) = 0 is necessary for a local extremum, but inflection points also satisfy f′ = 0 (e.g., f(x) = x³ at x = 0). The second derivative test resolves the ambiguity in one variable; the discriminant resolves it in two variables.