Maximum and minimum are known as the extrema of a function. They can be called as local or relative extrema that exist as highest or lowest function value within a part of domain of the function. If it is the highest or smallest function value (y-value) for “all x” in the domain of f, then it is known as absolute extrema. The maximum represents the largest value of the function or highest y-value at the peak of a curve (looks like a hill top). While the minimum represent the smallest value of the function at the bottom of the curve (looks like a valley). By visual inspection, the blue and yellow curves shows minimum and the red curve shows a maximum.
Aside from visual inspection, we can evaluate the maximum and minimum based on the following conditions.
Definition of Maximum and Minimum:
The function of f has a maximum at “c” that exist within the interval of (x1,x2) implies f(c) ≥ f(x).
The function of f has a minimum at “c” that exist within the interval of (x1,x2) implies f(c) ≤ f(x)
Note “c” represent a critical x-value in the function of f.
First derivative test:
For function of f that has a critical number “c” within a continuous and open interval, the f(c) can be classified as:
If f’(x) changes from negative to positive at “c”, then f(c) is a relative minimum.
If f’(x) changes from positive to negative at “c”, then f(c) is a relative minimum.
If f’(x) does not change sign at c, then f(c) is not a relative extrema.
Relative minimum: f’(x) <0 at the left side of c and f’(x)>0 at the right side of c.
Relative maximum: f’(x) >0 at the left side of c and f’(x)<0 at the right side of c.
Second Derivative test:
For a function that has f’(c) =0, the second derivative can be used to test if c is at the relative maximum or relative minimum.
If f”(c) >0, then f(c) is a relative minimum.
If f”(c)<0, the f(c) is a relative maximum.
In more simpler terms, when a parabola is shaped like a cup (concave up), it has a minimum.
When something is shaped like a frown (concave down), it has a max.
So yellow and blue are like cups, they have a minimum.
The red is a frown, it has a max.