Optimization Notes

Key Definitions

• Local Maximum and Minimum:

  • A function has a local maximum at x = c if f(c) ≥ f(x) for all x near c (akin to the highest point of a hill).

  • A function has a local minimum at x = c if f(c) ≤ f(x) for all x near c (similar to standing at the bottom of a valley).

• Global Maximum and Minimum:

  • Extend local definitions to the entire domain of the function, determining the highest/lowest point across the entire graph. Practical Examples

• Conceptual exercise: determine the number of local maxima for a given function between intervals A and B.

• Discussed reasons for determining local maxima, including endpoint values qualifying as local extrema.

Distance Optimization Problem

Setting Up the Problem

• Goal: optimize the distance between the point (0.75) and the line described by the equation y = 6 - 3x.

• Students to determine the distance between this point and another point on the line.

• Reviewed distance formula:

  • Distance between two points (x1, y1) and (x2, y2):[ \text{Distance} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} ]

• Given point on line as (x, 6 - 3x); students set up the function for distance and prepared to find critical points by taking derivatives.

Utilizing Derivatives

• Important note: simplification of the derivative calculation due to square root function being monotonically increasing.

• Critical points of the function with and without the square root are the same, streamlining the optimization process.