Extrema and the First-Derivative Test

Definition of Relative Extrema:
- $f(c)$ is a relative maximum if there exists an interval $(a, b)$ containing $c$ such that $f(x) \le f(c)$ for all $x$ in $(a, b)$ .
- $f(c)$ is a relative minimum if there exists an interval $(a, b)$ containing $c$ such that $f(x) \ge f(c)$ for all $x$ in $(a, b)$ .
Critical Numbers: Relative extrema occur at a critical number $c$ , where either $f'(c) = 0$ or $f'(c)$ is undefined.

Functionality: This test classifies critical numbers as relative minima, relative maxima, or neither based on the sign of the first derivative.
Classification Criteria:
- Relative Minimum: $f'(x)$ changes from negative to positive at $c$ .
- Relative Maximum: $f'(x)$ changes from positive to negative at $c$ .
- Neither: $f'(x)$ is positive on both sides or negative on both sides of $c$ .
Guidelines for Finding Relative Extrema:
1. Find the derivative $f'(x)$ .
2. Identify critical numbers.
3. Test the sign of $f'(x)$ in the intervals created by these numbers.
4. Use the test results to determine the nature of each critical point.

Concept: Relative extrema describe local behavior, while absolute extrema (global behavior) represent the single highest or lowest points on an entire interval.
Extreme Value Theorem: A continuous function on a closed interval $[a, b]$ must have both an absolute minimum and an absolute maximum.
Guidelines for Finding Extrema on $[a, b]$ :
1. Find critical numbers of $f$ in the open interval $(a, b)$ .
2. Evaluate $f$ at each critical number.
3. Evaluate $f$ at the endpoints $a$ and $b$ .
4. The smallest resulting value is the absolute minimum; the largest is the absolute maximum.

Scenario: Maximizing profit for a fast-food restaurant based on hamburger sales level $x$ .
Profit Function: $P = 2.44x - \frac{x^2}{20,000} - 5000$
Marginal Profit: Found by taking the derivative and setting it to zero: $2.44 - \frac{x}{10,000} = 0$
Critical Number: Selling $x = 24,400$ hamburgers yields the maximum profit.
Maximum Profit Value: Substituting the critical number into the original function results in a maximum profit of $\$24,768$ .