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Activity9.2Local Extrema

Referring back to Exercises 9.1.1, the vertex of the parabola in Exercise is called a local maximum point and the points \(\point{2}{C_3}\) and \(\point{2}{C_4}\) in Exercise are called local minimum points. Collectively, these points are called local extreme points.

While working through Exercises 9.1.1 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 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.

Definition9.2.1Critical 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.



The function \(g\) shown in Figure 9.2.2 has a vertical tangent line at \(-3\text{.}\)

Veronica says that \(-3\) is a critical number of \(g\) but Tito disagrees. Tito contends that \(-3\) is not a critical number because \(g\) does not have a local extreme point at \(-3\text{.}\) Who is right?


Answer each of the following questions (using complete sentences) in reference to the function \(f\) shown in Figure 9.2.3.


What are the critical numbers of \(f\text{?}\)


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.


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.

Decide whether each of the following statements is True or False.


True or False? A function always has a local extreme point at each of its critical numbers.


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 \(h\text{.}\)


True or False? If \(\fe{\fd{g}}{2.7}=0\text{,}\) then \(g\) must have a local extreme point at \(2.7\text{.}\)


True or False? If \(\fe{\fd{g}}{2.7}=0\text{,}\) then \(2.7\) must be a critical number of \(g\text{.}\)


True or False? If \(\fe{\fd{g}}{9}\) does not exist, then \(9\) must be a critical number of \(g\text{.}\)