Finding the highest and lowest points of a function is crucial in calculus. Absolute extrema are the overall max and min values, while relative extrema are local peaks and valleys. These concepts help us understand function behavior and solve real-world optimization problems.

The Extreme Value Theorem guarantees that continuous functions on closed intervals have absolute extrema. This powerful tool simplifies our search for max and min values, focusing on critical points and endpoints. It's essential for solving optimization problems in various fields.

Absolute and Relative Extrema

Defining Absolute and Relative Extrema

• Absolute maximum represents the largest value of a function $f(x)$ over its entire domain
  • Occurs at the highest point on the graph of the function
  • For $f(x) = -x^2 + 4x - 3$ on the interval $[0, 4]$, the absolute maximum is 1 at $x = 2$
• Absolute minimum signifies the smallest value of a function $f(x)$ over its entire domain
  • Occurs at the lowest point on the graph of the function
  • For $f(x) = x^2 - 4x + 5$ on the interval $[-1, 3]$, the absolute minimum is 1 at $x = 2$
• Relative maximum (local maximum) is the largest value of a function within a specific neighborhood or interval
  • Occurs at a peak or highest point within a localized region on the graph
  • For $f(x) = x^3 - 3x^2 - 9x + 7$, a relative maximum occurs at $x = -1$
• Relative minimum (local minimum) is the smallest value of a function within a specific neighborhood or interval
  • Occurs at a valley or lowest point within a localized region on the graph
  • For $f(x) = x^3 - 3x^2 - 9x + 7$, a relative minimum occurs at $x = 3$

Finding Absolute and Relative Extrema

• To find absolute extrema, evaluate the function at all critical points and endpoints of the domain
  • Critical points are where the derivative $f'(x) = 0$ or where $f'(x)$ is undefined
  • Compare the function values at these points to determine the absolute maximum and minimum
• Relative extrema can be identified using the First Derivative Test
  • If $f'(x)$ changes from positive to negative at a critical point, it is a relative maximum
  • If $f'(x)$ changes from negative to positive at a critical point, it is a relative minimum
• Second Derivative Test can also be used to classify relative extrema
  • If $f''(x) < 0$ at a critical point, it is a relative maximum
  • If $f''(x) > 0$ at a critical point, it is a relative minimum
  • If $f''(x) = 0$, the test is inconclusive, and further analysis is needed

Extreme Value Theorem

Closed and Bounded Sets

• A set is closed if it contains all of its boundary points (endpoints)
  • For an interval, square brackets $[a, b]$ indicate a closed set, while parentheses $(a, b)$ indicate an open set
  • Example of a closed set: $[0, 5]$, which includes the endpoints 0 and 5
• A set is bounded if it has both an upper and lower bound
  • Upper bound is a value greater than or equal to all elements in the set
  • Lower bound is a value less than or equal to all elements in the set
  • Example of a bounded set: $[2, 7]$, where 2 is the lower bound and 7 is the upper bound

Extreme Value Theorem

• The Extreme Value Theorem states that if a function $f(x)$ is continuous on a closed and bounded interval $[a, b]$, then $f(x)$ attains an absolute maximum and an absolute minimum on $[a, b]$
  • Guarantees the existence of both absolute extrema for continuous functions on closed and bounded domains
  • Helps narrow down the search for absolute extrema to critical points and boundary points
• Continuity is a crucial condition for the Extreme Value Theorem
  • A function is continuous if it has no gaps, jumps, or asymptotes within the given interval
  • Continuous functions have the property that small changes in input lead to small changes in output

Boundary Points and Extrema

• Boundary points are the endpoints of a closed interval $[a, b]$
  • In the interval $[2, 6]$, the boundary points are 2 and 6
• When finding absolute extrema using the Extreme Value Theorem, it is essential to evaluate the function at boundary points in addition to critical points
  • Absolute extrema can occur at either critical points or boundary points
  • Example: For $f(x) = x^3 - 9x^2 + 24x - 16$ on $[0, 4]$, evaluate $f(x)$ at critical points and boundary points 0 and 4
• If a function is not continuous or the domain is not closed and bounded, the Extreme Value Theorem does not apply
  • In such cases, other methods like analyzing limits and asymptotic behavior may be necessary to determine the existence of absolute extrema