4.7 Maxima/Minima Problems

3 min read•june 24, 2024

can have where the rate of change is zero in all directions. These points are found by setting to zero. The helps classify them as local minima, maxima, or saddle points.

Finding involves evaluating a function at critical points and along domain boundaries. This process is crucial for solving real-world problems, where we maximize or minimize a quantity subject to specific .

Critical Points and Extrema of Multivariable Functions

Critical points of two-variable functions

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  • Points where the function's rate of change is zero in all directions simultaneously
  • Found by setting all partial derivatives equal to zero and solving the resulting system of equations
  • Example: For f(x,y)=x2+y2f(x, y) = x^2 + y^2, set fx=2x=0f_x = 2x = 0 and fy=2y=0f_y = 2y = 0 to find the critical point at (0,0)(0, 0)
  • The vector (∇f) is zero at critical points

Classification of critical points

  • Second derivative test determines the nature of a critical point (, , or )
  • Calculates the D=fxx(x,y)⋅fyy(x,y)−[fxy(x,y)]2D = f_{xx}(x, y) \cdot f_{yy}(x, y) - [f_{xy}(x, y)]^2 at the critical point
    • D>0D > 0 and fxx(x,y)>0f_{xx}(x, y) > 0 indicates a local minimum
    • D>0D > 0 and fxx(x,y)<0f_{xx}(x, y) < 0 indicates a local maximum
    • D<0D < 0 indicates a saddle point
    • D=0D = 0 is inconclusive and requires further investigation
  • Example: For f(x,y)=x2−y2f(x, y) = x^2 - y^2, the critical point (0,0)(0, 0) has D=−4D = -4, indicating a saddle point
  • The can be used to classify critical points in higher dimensions

Absolute extrema in two variables

  • Overall maximum and minimum values of a function within a
  • Found by evaluating the function at critical points and boundary points
  • Steps to find absolute :
    1. Find all critical points within the domain
    2. Evaluate the function at each critical point
    3. Parameterize the domain's boundary (line segments for rectangles, curves for other shapes)
    4. Find critical points on the boundary by substituting the into the function
    5. Evaluate the function at each boundary critical point
    6. Compare function values at all critical points to identify absolute maximum and minimum
  • Example: For f(x,y)=x+yf(x, y) = x + y on the domain 0≤x≤10 \leq x \leq 1 and 0≤y≤10 \leq y \leq 1, evaluate ff at the critical point (0,0)(0, 0) and the four boundary segments to find the absolute extrema

Optimization and Applications

Solving optimization problems

  • Involves finding the maximum or minimum value of a function subject to given constraints
  • Steps to solve optimization problems:
    1. Identify the (quantity to be maximized or minimized)
    2. Identify the constraints on the variables
    3. Express the objective function in terms of a single variable using the constraints
    4. Determine the domain based on the problem context
    5. Find the absolute extrema of the objective function on the given domain
    6. Interpret the results in the context of the original problem
  • Example: Minimizing the surface area of a cylindrical can with a fixed volume (constraint) by expressing surface area in terms of a single variable (radius or height) and finding the absolute minimum within the domain determined by the can's dimensions

Advanced Optimization Techniques

  • method for constrained optimization problems
  • Directional derivatives to analyze function behavior in specific directions
  • Gradient-based optimization algorithms for numerical solutions

Key Terms to Review (19)

Absolute Extrema: Absolute extrema refer to the maximum and minimum values that a function can attain over its entire domain. These represent the global or absolute maximum and minimum points of the function, as opposed to local extrema which are the highest and lowest points within a specific interval.
Closed and bounded domain: A closed and bounded domain refers to a set in which all points are contained within a finite region and include their boundary points. This concept is crucial for understanding the behavior of functions, particularly when determining where to find maximum and minimum values, as functions can achieve extrema only within such domains.
Constraints: Constraints are conditions or limitations that must be satisfied in optimization problems, often defining the feasible region where solutions can be found. They play a crucial role in determining the maximum or minimum values of a function by restricting the possible values that the variables can take. Understanding constraints helps in formulating the problem accurately and ensuring that any solutions derived are valid within the given limitations.
Critical Points: Critical points refer to the points in a function of several variables where the partial derivatives of the function vanish or become undefined. These points are of particular importance in the analysis of maxima, minima, and saddle points of a function, as well as in the application of Lagrange multipliers to constrained optimization problems.
Directional Derivative: The directional derivative is a measure of the rate of change of a function in a specific direction at a given point. It represents the slope of the function in a particular direction, providing information about how the function is changing along that direction.
Discriminant: The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It is a key concept in both the study of conic sections and in solving optimization problems involving maxima and minima.
Extrema: Extrema refer to the maximum and minimum values of a function within a specified domain. These points are crucial for understanding the behavior of functions, as they indicate where the function reaches its highest or lowest points, which is essential in optimization problems and analyzing the overall shape of the graph.
Gradient: The gradient is a vector that represents the direction and rate of the fastest increase of a scalar function. It provides essential information about how a function changes in space, connecting to concepts such as optimizing functions, understanding the behavior of multi-variable functions, and exploring the properties of vector fields.
Hessian Matrix: The Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. It is a fundamental tool in the analysis of critical points and optimization problems involving functions of multiple variables.
Lagrange Multipliers: Lagrange multipliers are a mathematical technique used to find the maximum or minimum value of a function subject to one or more constraints. This method allows for the optimization of a function while considering the limitations or restrictions imposed by the constraints.
Local maximum: A local maximum is a point in a function where the value of the function is greater than the values of the function at nearby points. It is a key concept when analyzing functions to determine their behavior, as local maxima indicate peaks in the graph. Understanding local maxima is crucial for optimization problems, where one seeks to find the highest or lowest points of a function over a specified interval.
Local Minimum: A local minimum is a point on a function's graph where the function value is less than or equal to the function values in the immediate surrounding area. It represents a point where the function attains a minimum value within a localized region, even if it is not the absolute minimum value of the function over its entire domain.
Multivariable Functions: A multivariable function is a function that depends on more than one independent variable. These functions are used to model complex real-world phenomena that cannot be adequately described by a single variable. They are central to the study of calculus of several variables, which extends the concepts of limits, continuity, differentiation, and optimization from functions of a single variable to functions of multiple variables.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, which can involve either maximizing or minimizing a quantity. In optimization, the objective function is typically subject to constraints that limit the feasible region for potential solutions. Understanding how to formulate and analyze an objective function is crucial for finding optimal solutions in various mathematical and real-world applications.
Optimization: Optimization is the process of making something as effective or functional as possible, often by finding the best solution to a problem under given constraints. In mathematics, this involves identifying the maximum or minimum values of a function, which is crucial for solving real-world problems like maximizing profits or minimizing costs. Understanding how to optimize functions involves utilizing techniques like finding critical points and applying the second derivative test.
Parameterization: Parameterization is the process of representing a mathematical object, such as a curve, surface, or higher-dimensional manifold, using a set of parameters or variables. This technique allows for a more flexible and convenient way to describe and analyze these objects, as it provides a way to express their properties and behaviors in terms of the underlying parameters.
Partial Derivatives: Partial derivatives are a type of derivative that measure the rate of change of a multivariable function with respect to one of its variables, while treating the other variables as constants. They provide a way to analyze the sensitivity of a function to changes in its individual inputs.
Saddle Point: A saddle point is a critical point of a function where the function has a local maximum in one direction and a local minimum in an orthogonal direction. It represents a point of equilibrium where the function neither strictly increases nor strictly decreases in all directions.
Second Derivative Test: The second derivative test is a method used to determine the critical points of a function and classify them as local maxima, local minima, or points of inflection. It relies on the sign of the second derivative to make this determination.
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