Critical points of multivariable functions are places where a surface can peak, dip, or change direction like a mountain pass. For a function z = f(x,y), these points help describe the shape of the graph and the behavior of systems with two inputs. They matter in physics, engineering, economics, and data science because many real problems involve optimizing a quantity that depends on more than one variable.
A critical point is often the first place to look for a maximum, minimum, or saddle point.
Key Facts
- A critical point of f(x,y) occurs where fx = 0 and fy = 0, or where one partial derivative does not exist.
- The gradient is ∇f = <fx, fy>, and critical points usually satisfy ∇f = <0, 0>.
- The Hessian matrix is H = [[fxx, fxy], [fyx, fyy]].
- For two variables, the discriminant is D = fxx fyy - (fxy)^2.
- If D > 0 and fxx > 0, the critical point is a local minimum; if D > 0 and fxx < 0, it is a local maximum.
- If D < 0, the critical point is a saddle point; if D = 0, the second derivative test is inconclusive.
Vocabulary
- Critical point
- A point in the domain where all first partial derivatives are zero or where at least one first partial derivative is undefined.
- Gradient
- The vector of first partial derivatives that points in the direction of steepest increase of a function.
- Hessian matrix
- A square matrix of second partial derivatives used to study the curvature of a multivariable function.
- Saddle point
- A critical point where the function curves upward in one direction and downward in another direction.
- Contour curve
- A curve in the xy-plane where a function f(x,y) has a constant value.
Common Mistakes to Avoid
- Solving only fx = 0 and forgetting fy = 0 is wrong because a critical point of f(x,y) requires both first partial derivatives to vanish, unless a derivative is undefined.
- Calling every critical point a maximum or minimum is wrong because saddle points also have zero gradient but are not extrema.
- Using D = fxx fyy - fxy instead of D = fxx fyy - (fxy)^2 is wrong because the mixed partial term must be squared in the two-variable Hessian test.
- Ignoring the case D = 0 is wrong because the second derivative test gives no conclusion, so another method such as analyzing nearby values is needed.
Practice Questions
- 1 Find and classify the critical point of f(x,y) = x^2 + y^2 - 4x + 6y.
- 2 Find and classify all critical points of f(x,y) = x^2 - y^2 + 2x + 4y.
- 3 A surface has a critical point at (a,b), and contour curves near the point look like hyperbolas crossing in opposite directions rather than nested closed loops. Explain why this suggests a saddle point instead of a local maximum or local minimum.