∬Differential Calculus Unit 13 – Maximum and Minimum Values

Key Concepts

• Maximum and minimum values represent the highest and lowest points on a function's graph
• Critical points are where the function's derivative is zero or undefined
• The First Derivative Test uses the sign of the derivative to determine whether a critical point is a local maximum, local minimum, or neither
• The Second Derivative Test uses the sign of the second derivative to classify critical points as local maxima, local minima, or neither
• Absolute extrema are the overall highest and lowest points on a function's graph within a given interval
  • Absolute maximum is the highest point on the graph
  • Absolute minimum is the lowest point on the graph
• Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints
• Identifying the domain of a function is crucial for finding absolute extrema and solving optimization problems

Finding Critical Points

• Critical points occur where the derivative of a function is zero or undefined
• To find critical points, set the first derivative equal to zero and solve for the input variable
  • For example, if $f(x) = x^3 - 3x^2 - 9x + 1$, then $f'(x) = 3x^2 - 6x - 9$
  • Set $f'(x) = 0$ and solve for $x$ to find the critical points
• Check for any values that make the derivative undefined, such as denominators equal to zero or expressions under even roots equal to negative numbers
• Classify the critical points as local maxima, local minima, or neither using the First or Second Derivative Test
• Remember that endpoints of a closed interval can also be critical points when finding absolute extrema

First Derivative Test

• The First Derivative Test determines the nature of critical points by examining the sign of the derivative on either side of the point
• If the derivative changes from positive to negative at a critical point, it is a local maximum
• If the derivative changes from negative to positive at a critical point, it is a local minimum
• If the derivative does not change sign at a critical point, it is neither a local maximum nor a local minimum
  • This can occur at inflection points or saddle points
• To apply the First Derivative Test:
  1. Find the critical points of the function
  2. Evaluate the sign of the derivative on either side of each critical point
  3. Classify the critical points based on the sign change

Second Derivative Test

• The Second Derivative Test determines the nature of critical points using the sign of the second derivative
• If the second derivative is negative at a critical point, it is a local maximum
• If the second derivative is positive at a critical point, it 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
• To apply the Second Derivative Test:
  1. Find the critical points of the function
  2. Calculate the second derivative of the function
  3. Evaluate the second derivative at each critical point
  4. Classify the critical points based on the sign of the second derivative

Absolute Extrema

• Absolute extrema are the overall maximum and minimum values of a function within a given interval
• To find absolute extrema on a closed interval $[a, b]$:
  1. Find the critical points of the function within the interval
  2. Evaluate the function at each critical point and the endpoints of the interval
  3. Compare the function values to determine the absolute maximum and minimum
• If the interval is open or unbounded, the function may not have absolute extrema
  • In this case, consider the limit of the function as the input approaches infinity or the endpoint(s) of the interval
• Absolute extrema can occur at critical points or endpoints of the interval

Optimization Problems

• Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints
• To solve an optimization problem:
  1. Identify the objective function (the quantity to be maximized or minimized)
  2. Identify the constraint(s) on the variables
  3. Use the constraint(s) to express the objective function in terms of a single variable
  4. Find the critical points of the objective function
  5. Evaluate the objective function at the critical points and any relevant endpoints
  6. Determine the maximum or minimum value and the corresponding input value(s)
• Common types of optimization problems include:
  • Maximizing profit or minimizing cost
  • Maximizing area or volume subject to perimeter or surface area constraints
  • Minimizing distance or time traveled

Common Pitfalls

• Forgetting to consider the endpoints of a closed interval when finding absolute extrema
• Misclassifying critical points due to sign errors when applying the First Derivative Test
• Assuming the Second Derivative Test always provides conclusive results
  • Remember that if the second derivative is zero, the test is inconclusive
• Failing to identify all critical points, especially those where the derivative is undefined
• Incorrectly setting up the objective function or constraints in optimization problems
• Not checking the domain of the function to ensure critical points and absolute extrema are valid
• Confusing local extrema with absolute extrema

Real-World Applications

• Optimization techniques are used in various fields, such as economics, engineering, and physics
• In business, optimization can be used to maximize profit or minimize costs by finding the optimal production levels or pricing strategies
• Engineers use optimization to design structures, machines, and systems that maximize efficiency or minimize material usage
  • For example, designing a container with minimal surface area to hold a given volume
• In physics, optimization can be used to determine the path of least action or the shortest distance between two points
  • Fermat's Principle states that light follows the path of least time between two points
• Transportation and logistics companies use optimization to plan the most efficient routes for deliveries or minimize fuel consumption
• Optimization is also used in machine learning and artificial intelligence to find the best model parameters that minimize prediction errors