∞Calculus IV Unit 7 – Maximum and Minimum Values

Key Concepts and Definitions

• Maximum and minimum values represent the highest and lowest points on a function's graph
• Local (relative) extrema occur at critical points where the function changes from increasing to decreasing or vice versa
• Global (absolute) extrema are the overall highest and lowest points on a function's graph within a given domain
• Critical points are values of x where the derivative of the function is either zero or undefined
• The first derivative test determines the nature of a critical point by examining the sign of the derivative on either side of the point
• The second derivative test determines the nature of a critical point by evaluating the second derivative at that point
  • If the second derivative is negative, the point is a local maximum
  • If the second derivative is positive, the point is a local minimum
• Optimization problems involve finding the maximum or minimum value of a function subject to given constraints

Finding Critical Points

• To find critical points, set the first derivative of the function equal to zero and solve for x
• Also, consider values of x where the first derivative is undefined (such as when the denominator of a rational function is zero)
• Example: For the function $f(x) = x^3 - 3x^2 - 9x + 1$, set $f'(x) = 3x^2 - 6x - 9 = 0$ and solve for x
  • This yields critical points at $x = 3$ and $x = -1$
• Check the endpoints of the domain if the function is defined on a closed interval
• Evaluate the function at each critical point and endpoint to determine the absolute maximum and minimum values
• Be cautious of critical points that occur at discontinuities or corners of the function's graph
• Remember that a function may have multiple local extrema but only one absolute maximum and one absolute minimum on a given domain

First Derivative Test

• The first derivative test examines the sign of the derivative on either side of a critical point to determine its nature
• If the derivative changes from positive to negative at a critical point, the point is a local maximum
• If the derivative changes from negative to positive at a critical point, the point is a local minimum
• If the derivative has the same sign on both sides of a critical point, the point is neither a maximum nor a minimum (known as a saddle point)
• Example: Consider the function $f(x) = x^3 - 3x^2 - 9x + 1$ with critical points at $x = 3$ and $x = -1$
  • At $x = 3$, $f'(x)$ changes from positive to negative, indicating a local maximum
  • At $x = -1$, $f'(x)$ changes from negative to positive, indicating a local minimum
• The first derivative test is useful when the second derivative test is inconclusive (i.e., when the second derivative is zero at a critical point)

Second Derivative Test

• The second derivative test evaluates the second derivative at a critical point to determine its nature
• If the second derivative is negative at a critical point, the point is a local maximum
• If the second derivative is positive at a critical point, the point is a local minimum
• If the second derivative is zero at a critical point, the test is inconclusive, and the first derivative test should be used instead
• Example: For the function $f(x) = x^4 - 4x^3 + 4x^2$, the critical points are $x = 0$ and $x = 2$
  • At $x = 0$, $f''(0) = 8 > 0$, indicating a local minimum
  • At $x = 2$, $f''(2) = -8 < 0$, indicating a local maximum
• The second derivative test is often quicker and easier to apply than the first derivative test, but it may not always provide a conclusive result

Absolute vs. Relative Extrema

• Absolute (global) extrema are the overall highest and lowest points on a function's graph within a given domain
• Relative (local) extrema occur at critical points where the function changes from increasing to decreasing or vice versa
• A function may have multiple relative extrema but only one absolute maximum and one absolute minimum on a given domain
• To find absolute extrema on a closed interval, evaluate the function at all critical points and endpoints, then compare the values
• Example: Consider the function $f(x) = x^3 - 3x^2 - 9x + 1$ on the interval $[-2, 4]$
  • The critical points are $x = 3$ (local maximum) and $x = -1$ (local minimum)
  • Evaluating the function at the critical points and endpoints: $f(-2) = -37$, $f(-1) = 12$, $f(3) = -44$, $f(4) = -31$
  • The absolute maximum is $f(-1) = 12$, and the absolute minimum is $f(3) = -44$
• Functions defined on open intervals may not have absolute extrema, as the function may approach infinity or negative infinity at the endpoints

Optimization Problems

• Optimization problems involve finding the maximum or minimum value of a function subject to given constraints
• Identify the quantity to be maximized or minimized (the objective function) and express it in terms of the relevant variables
• Determine any constraints on the variables and use them to eliminate variables or establish relationships between them
• Find the critical points of the objective function, considering both interior points and boundary points (where constraints are active)
• Evaluate the objective function at each critical point and compare the values to determine the maximum or minimum
• Example: Maximize the volume of a rectangular box with a square base and a fixed surface area of 108 square units
  • Let x be the side length of the square base and h be the height. The volume is $V = x^2h$
  • The surface area constraint is $2x^2 + 4xh = 108$, which can be rewritten as $h = (108 - 2x^2) / 4x$
  • Substitute the constraint into the volume formula: $V = x^2(108 - 2x^2) / 4x = 27x - x^3/2$
  • Find the critical points by setting $V' = 27 - 3x^2/2 = 0$, which yields $x = 6$ (the only positive solution)
  • Evaluate the volume at $x = 6$: $V(6) = 27(6) - 6^3/2 = 54$, the maximum volume
• When solving optimization problems, be sure to consider any implicit constraints (e.g., non-negative dimensions) and check the endpoints of the domain

Multivariable Extrema

• Multivariable functions have input variables in two or more dimensions, such as $f(x, y)$ or $f(x, y, z)$
• To find critical points of a multivariable function, set the partial derivatives equal to zero and solve the resulting system of equations
• The second derivative test for multivariable functions involves the Hessian matrix, which contains the second-order partial derivatives
• If the Hessian is positive definite at a critical point, the point is a local minimum; if it is negative definite, the point is a local maximum
• Saddle points occur when the Hessian has both positive and negative eigenvalues at a critical point
• Example: Consider the function $f(x, y) = x^2 + y^2 - 2x - 4y + 2$
  • Set $f_x = 2x - 2 = 0$ and $f_y = 2y - 4 = 0$ to find the critical point at $(1, 2)$
  • The Hessian matrix at $(1, 2)$ is $\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, which is positive definite, indicating a local minimum
• Constrained optimization problems in multiple variables can be solved using the method of Lagrange multipliers

Applications in Real-World Scenarios

• Optimization techniques are used in various fields, such as engineering, economics, and physics, to make the best decisions under given constraints
• In business, optimization can help maximize profits, minimize costs, or optimize resource allocation
  • Example: A company wants to minimize the cost of producing a product while meeting customer demand and production constraints
• In engineering, optimization is used to design efficient structures, machines, or systems
  • Example: Designing a bridge to minimize material cost while ensuring sufficient strength and stability
• In physics, optimization principles often appear in the form of variational problems or the principle of least action
  • Example: Fermat's principle states that light travels along the path that minimizes the time taken between two points
• Optimization is crucial in machine learning and artificial intelligence for training models and minimizing loss functions
  • Example: Gradient descent is an optimization algorithm used to minimize the cost function in neural networks
• Transportation and logistics rely on optimization to plan efficient routes, schedules, and resource allocation
  • Example: Finding the shortest path for a delivery truck to visit multiple locations while considering traffic, time windows, and vehicle capacity
• Optimization techniques are essential in finance for portfolio management, risk assessment, and trading strategies
  • Example: Maximizing expected returns while minimizing risk in a portfolio of investments