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.
Understanding Calculus: Critical Points of Multivariable Functions
Start by treating each input separately. The partial derivative with respect to x measures how the output changes when x moves while y is held fixed. The partial derivative with respect to y does the same in the other horizontal direction.
Solving the two resulting equations together finds candidate locations. This is a simultaneous equation problem, so every solution must satisfy both conditions at once. Check the original function domain too.
A formula may produce a candidate where a denominator is zero or where a square root is not allowed. Such a point is not part of the surface.
The next task is to inspect the surface near each candidate, not just at the candidate itself. Second partial derivatives describe curvature. Curving upward in the x direction suggests a bowl along that slice.
Curving downward suggests an upside down bowl. The mixed partial derivative tracks how the slope in one direction changes as the other input changes. Together, these values show whether nearby motion raises the function, lowers it, or does different things in different directions.
A saddle point can rise when moving one way yet fall when moving another way. It is therefore not a highest or lowest nearby value, even though the slope is flat at the center.
The second derivative test is a quick local test, not a complete answer to every optimization problem. When its discriminant is zero, the surface may be unusually flat and more investigation is needed. Substitute nearby points, examine slices, or use higher derivatives.
For example, a function equal to x to the fourth power plus y to the fourth power has a very flat local minimum at the origin. A function equal to x to the fourth power minus y to the fourth power has a saddle point there. Both examples need more than the usual second derivative conclusion.
If a problem asks for an absolute maximum or minimum on a closed region, inspect the boundary as well. The best value can occur along an edge or at a corner rather than at an interior critical point.
These ideas appear whenever a result depends on two adjustable quantities. An engineer might vary the width and thickness of a part to reduce material use while keeping it strong. A business model can estimate cost from price and production level.
In physics, potential energy can depend on position in two directions. A stable equilibrium behaves like a local minimum because small displacements increase energy. An unstable balance behaves more like a local maximum or a saddle.
When learning, sketch level curves before trusting algebra. Closed curves around a point often suggest a peak or dip.
Curves that cross or change orientation can signal a saddle. Keep local and absolute results separate, and always state which kind of extremum the calculation supports.
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.