Chapter 4, Problem 1RE

For Exercises 1–2, indicate all critical points on the given graphs. Which correspond to local minima, local maxima, global maxima, global minima, or none of these? (Note that the graphs are on closed intervals.)

To determine

To indicate: All critical points on the given graphs, which correspond to local maxima, local minima, global maxima, global minima or none of these.

Explanation of Solution

Definition used:

Local maxima and local minima:

Suppose p is a point in the domain of f(x):

1. f(x) has a local minimum at p if f(p) is less than or equal to the values of f(x) for points near p.

2. f(x) has a local maximum at p if f(p) is greater than or equal to the values of f(x) for points near p.

Global maxima and minima:

Suppose p is a point in the domain of f(x):

1. f(x) has a global minimum at p if f(p) is less than or equal to all values of f(x).

2. f(x) has a global maximum at p if f(p) is greater than or equal to all values of f(x).

Theorem used:

If f is continuous on the closed interval $a\le x\le b$, then f has a global maximum and a global minimum on that interval.

Calculation:

It is given that, the graph is on closed interval, by the above theorem the function will have global maximum and a global minimum on the interval.

Redraw the graphs marking the critical points as shown below in Figure 1.

From Figure 1, it is notices that all critical points on the graph is showed.