notes/education/math/MATH1210 (calc 1)/Max and Min.md

Maximum/Minimum

A function f has an absolute maximum at c if f(c) >= f(x). We call f(c) the maximum value of f. The absolute maximum is the largest possible output value for a function.

A function f has an absolute minimum at c if f(c) <= f(x). f(c) is the absolute minimum value of f. The absolute minimum is the smallest possible output value for a function.

• Where the derivative of a function is zero, there is either a peak or a trough.

Critical Numbers

A number is considered critical if the output of a function exists and \dfrac{d}{dx} is zero or undefined.

Local Max/Min

A local max/min is a peak or trough at any point along the graph.

Extreme Value Theorem

If f is a continuous function in a closed interval [a, b], then f achieves both an absolute maximum and an absolute minimum in [a, b]. Furthermore, the absolute extrema occur at a or at b or at a critical number between a and b.

Example

Find the absolute maximum and absolute minimum of the function f(x) = x^2 -3x + 2 on the closed interval [0, 2]:

1. x=0 and x=2 are both critical numbers because they are endpoints. Endpoints are always critical numbers because \dfrac{d}{dx} is undefined.
2. \dfrac{d}{dx} x^2 -3x + 2 = 2x -3
3. Setting the derivative to zero, 0 = 2x - 3
4. Solving for x, we get x = \dfrac{3}{2}. Three halves is a critical number because $