1. MTH1170 Function Extrema

2. Preliminary
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

3. Increasing & Decreasing Functions
Examine the function graphed above. Where is this function increasing, and where is this function decreasing?

4. Increasing & Decreasing Functions
It is easy to visually determine this when given the graph of the function, but how can we determine it mathematically when only the function is provided?

5. Increasing & Decreasing Functions
This implies that:
f’(x) > 0 when the function is increasing
f’(x) < 0 when the function is decreasing
f’(x) = 0 between increasing and decreasing regions (critical points)

6. The First Derivative Test
The Process: 1. Differentiate the function and solve for y’ if required. 2. Find the zeros of the derivative. 3. Place the zeros of the derivative on a number line and test to see if the derivative is positive or negative in the regions in between the zeros. 4. Using this information we can see where the function is increasing or decreasing.

7. Extreme Values (Maxima & Minima)
A function will have Extreme Values over its domain (either a specified domain, or wherever it is defined) of four different types: 1. Absolute Maximum - This is the greatest value f(x) obtains on its domain. 2. Absolute Minimum - This is the lowest value f(x) obtains on its domain. 3. Local Maximum - This is the greatest value f(x) obtains within the nearby vicinity. 4. Local Minimum - This is the lowest value f(x) obtains within the nearby vicinity.

8. Example
Determine the Extreme Values for the graphed function f(x).

9. How to Find Extreme Values
A function can have Extreme Values only at points of three special types:
1) Critical Points - f’(x) = 0
2) Singular Points - f’(x) is not defined
3) Endpoints in the domain of f(x)
To find Extreme Values we first differentiate f(x), then numerically find the critical points, singular points, and endpoints. After we have the x coordinates of these points, we evaluate the original function f(x) and compare the results against each other. 

10. Example
Find the maximum and minimum values of the following function over the specified domain.

11. Example
Find the max and min values of f(x) on the interval provided.