Max and Min Values of Functions

Understanding Maximum and Minimum Values in Functions

Maximum and Minimum Value Locations

  • Definition of Maximum Value: The value of a function at point $b$ such that $f(b) \geq f(x)$ for all $x$ in the domain of the function.
  • Definition of Minimum Value: The value of a function at point $a$ such that $f(a) \leq f(x)$ for all $x$ in the domain.
  • Endpoints Consideration:
    • Maximum or minimum values can occur at endpoint values of the function's interval (denoted as $x = a$ and $x = b$).
    • Example: Minimum can occur at an endpoint, maximum at the other endpoint.

Interior Points Consideration

  • Functions may exhibit maximums and minimums at points other than endpoints.
  • Key Point: For a minimum to occur at a point, the values of the function at all points nearby must be greater.
  • If a maximum happens at a point, values of the function must reduce toward either side of that maximum point.

Derivative and Critical Points

  • Tangent Line Concept:
    • At maximum or minimum points, the derivative (slope of the tangent line) must equal zero.
    • If the tangent line is flat, it indicates a point where the derivative is zero.
  • **Critical Points Definition: ** Places where the derivative is either zero or undefined.
  • If the point is the minimum/maximum, the derivative should be:
    • For Minimum: flat (zero slope)
    • For Maximum: flat (zero slope)

Cases of Undefined Derivative

  • A critical point may also occur where the derivative does not exist (e.g., sharp corn or cusp).
    • Example: A minimum can occur at a sharp corner without an existing derivative at that point.

Summary of Points to Check for Extremes

  • Endpoints: Check the values at both endpoints ($x = a$ and $x = b$).
  • Flat Derivative Points: Check the points where the derivative equals zero.
  • Undefined Derivative Points: Points where the derivative does not exist (e.g., cusps).

Complete List of Locations for Maximum and Minimum Values

  • Maximum and minimum values occur at:
    • Endpoints of the interval.
    • At points within the interval where the derivative equals zero.
    • At points within the interval where the derivative is undefined.

Why These Points are Sufficient for Finding Extrema

  • If a point is not an endpoint and the derivative is not zero or undefined, then it cannot be a maximum or minimum.
  • Argument Logic:
    1. If the derivative is positive, the function is increasing, thus not at maximum.
    2. If the derivative is negative, the function is decreasing, thus not at maximum.
  • Therefore, for a maximum to occur away from endpoints, the derivative must be zero.

Practical Steps for Finding Maximums and Minimums

  1. Identify the interval: Define a closed interval [a, b].

    • Example: from $x = -2$ to $x = 4$.

  2. Check the endpoints: Evaluate the function at the endpoints.

    • For $f(-2)$, $f(4)$, evaluate specific function values.

  3. Determine Critical Points: Find derivative.

    • Example: Given $f(x) = 2x^3 - 6x^2 - 18x + 4$, compute the derivative:
      • $f'(x) = 6x^2 - 12x - 18$.
    • Set $f'(x) = 0$ and solve for $x$.
    • Identify critical numbers:
      • $x = -1$ and $x = 3$.
    • Note: No points where the derivative is undefined since it's a polynomial.

  4. Evaluate function at critical points: Plug back into the original function.

    • Find $f(-1)$, $f(3)$ to compare with endpoints.

  5. Determine extrema:

    • Example results show:
      • At $x = -1$, $f(-1) = -14$ (minimum).
      • At $x = 3$, $f(3) = -50$ (note: potential maximum status).
    • At endpoints, compute:
      • $f(-2) = 0$
      • $f(4) = -36$.

  6. Conclusion: Identify maximum and minimum found among evaluated points.

    • For maximum, select highest output; for minimum, select lowest output.
    • In this case:
      • Maximum at $x = -1$ with value $14$.
      • Minimum at $x = 3$ with value $-50$.

Summary of Techniques

  • Always check endpoints, critical points (where the derivative is zero or undefined), and use calculated values to ascertain maximum and minimum values systematically.