Definition 9.2.1 Critical Numbers.
If \(f\) is a function, then we define the critical numbers of \(f\) as the numbers in the domain of \(f\) where the value of \(\fd{f}\) is either zero or does not exist.
Activity 9.2 Local Extrema
Referring back to Exercises 9.1, the vertex of the parabola in Exercise 9.1.1 is called a local maximum point and the points \(\point{2}{C_3}\) and \(\point{2}{C_4}\) in Exercise 9.1.2 are called local minimum points. Collectively, these points are called local extreme points.
While working through Exercises 9.1 you hopefully came to the conclusion that the local extreme points had certain characteristics in common. In the first place, they must occur at a number in the domain of the function (which eliminated \(f_1\) and \(f_2\) from contention in Exercise 9.1.2). Secondly, one of two things must be true about the first derivative when a function has a local extreme point: it either has a value of zero or it does not exist. This leads us to the definition of a critical number of a function.
Exercises Exercises
1.
Exercise Group.
2.
What are the critical numbers of \(f\text{?}\)
3.
What are the local extreme points on \(f\text{?}\) Classify the points as local minimums or local maximums and remember that points on the plane are represented by ordered pairs.
4.
Find the absolute maximum value of \(f\) over the interval \(\ointerval{-7}{7}\text{.}\) Please note that the function value is the value of the \(y\)-coordinate at the point on the curve and as such is a number.
Exercise Group.
Decide whether each of the following statements is True or False.
5.
True or False? A function always has a local extreme point at each of its critical numbers.
6.
True or False? If the point \(\point{t_1}{\fe{h}{t_1}}\) is a local minimum point on \(h\text{,}\) then \(t_1\) must be a critical number of \(h\text{.}\)
7.
True or False? If \(\fe{\fd{g}}{2.7} = 0\text{,}\) then \(g\) must have a local extreme point at \(2.7\text{.}\)
8.
True or False? If \(\fe{\fd{g}}{2.7} = 0\text{,}\) then \(2.7\) must be a critical number of \(g\text{.}\)
9.
True or False? If \(\fe{\fd{g}}{9}\) does not exist, then \(9\) must be a critical number of \(g\text{.}\)
