Name: finding critical points in multivariable calculus
Uploaded: 2026-06-08T09:38:57.661Z
Duration: 1 min 31 s
Channel: JoVE

A function of two variables can reach maximum or minimum values at specific points on its surface.A local maximum appears when the function value is higher than all nearby points, while a local minimum appears when it is lower.These extremes happen at critical points, where the tangent plane is locally flat.To find them, calculate the partial derivatives with respect to both variables. A point is critical when both derivatives are zero, indicating no change in any direction.For example, imagine 12 square meters of cardboard is given to build the largest possible open-top box. While the volume depends on length x, width y, and height z, the limited material acts as a constraint.This constraint allows height z to be expressed as a function of x and y, reducing the volume formula to just two variables.The goal is to find the maximum volume, which occurs at a critical point where the volume function is locally flat.To locate this point, the partial derivatives of V on x and y are calculated and set to zero. Solving these resulting equations gives the length and width. Substituting these values into the volume function gives the maximum volume.

local maximum and minimum values

In multivariable calculus, a function of two variables can exhibit local maximum or minimum values at certain points on its surface. A local maximum occurs when the function&#x27;s value at a point is greater than at all nearby points, while a local minimum occurs when the function&#x2019;s value is less than at all nearby locations. These points are referred to as local extrema and are of central importance in optimization problems.Local extrema are found at critical points, where the surface becomes momentarily flat in all directions. At such points, the rate of change of the function with respect to each variable&#x2014;called the partial derivatives&#x2014;is zero. A point is considered critical if both the partial derivative with respect to the first variable and the partial derivative with respect to the second variable are equal to zero. This means that small movements in either direction do not cause the function&#x27;s value to change.To determine whether a critical point represents a maximum, minimum, or saddle point, additional methods such as examining the second-order partial derivatives or using the Hessian matrix are required.As an application, consider maximizing the volume of an open-top rectangular box constructed from 12 square meters of cardboard. The box&#x2019;s volume depends on three dimensions&#x2014;length, width, and height&#x2014;but the total surface area provides a constraint. By expressing volume in terms of two variables, such as length and width, and incorporating the constraint into the model, one can apply optimization techniques.Partial derivatives of the resulting volume function with respect to each variable are calculated and set to zero to locate the critical points. Solving these conditions yields the box dimensions that provide the maximum volume. At the critical point, the function attains a local maximum, and the box reaches an optimal volume of 4 cubic meters. This process illustrates how the concept of critical points is used to solve practical optimization problems involving multiple variables and constraints.

Local Maximum and Minimum Values

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JoVE Core

Calculus

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Partial Derivatives and Gradients

KEY TERMS AND CONCEPTS

finding critical points in multivariable calculus

Multivariable calculus uses critical points to locate local maximum and minimum values on a surface. A local maximum is a point where the function value is higher than at nearby points. A local minimum is a point where the function value is lower than at nearby points. These points are called local extrema, and they matter in optimization problems.

Local extrema occur at critical points, where the surface is momentarily flat in all directions. At a critical point, the partial derivatives are zero. That means the rate of change with respect to the first variable is zero, and the rate of change with respect to the second variable is also zero. Small moves in either direction do not change the function value at that point.

A critical point does not always mean a maximum or minimum. It can also be a saddle point. To tell the difference, mathematicians use second-order partial derivatives or the Hessian matrix. These tools help identify whether the point is a local maximum, a local minimum, or neither.

The transcript also shows how this idea works in an optimization problem. It asks how to maximize the volume of an open-top rectangular box made from 12 square meters of cardboard. The volume depends on length, width, and height, but the surface area gives a constraint. By rewriting the volume in terms of two variables, such as length and width, the problem becomes easier to solve.

After the constraint is built into the model, partial derivatives of the volume function are found and set equal to zero. Solving those equations gives the critical point for the box. That point produces the maximum volume. The best box has a volume of 4 cubic meters, which shows how critical points can solve real optimization problems with multiple variables.

Finding Critical Points in Multivariable Calculus

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Learn how critical points, partial derivatives, and the Hessian matrix help find local maxima and minima, with a box-volume optimization example.

Finding Critical Points in Multivariable Calculus | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Critical Points in Multivariable Calculus

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