Question

4.3 max & min on an interval

1. a. find the (x,y) coordinates of all local minimums of w(x).

b. find the(x,y) coordinates of all local maximums of w(x).
c. does w(x) have an absolute minimum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this minimum.
d. does w(x) have an absolute maximum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this maximum.
e. find the (x,y) coordinates of the absolute maximum(s) of (x,y) on -1,1.
f. find the (x,y) coordinates of the absolute minimum(s) of (x,y) on -1,1.
g. find the (x,y) coordinates of the absolute minimum(s) of (x,y)

Explanation:

Step1: Recall local - minimum definition

A local minimum occurs where the function changes from decreasing to increasing.

Step2: Analyze the graph

By observing the graph of \(y = w(x)\), we look for points where the curve bottoms - out.

Step1: Recall local - maximum definition

A local maximum occurs where the function changes from increasing to decreasing.

Step2: Analyze the graph

By observing the graph of \(y = w(x)\), we look for points where the curve peaks.

Step1: Recall absolute - minimum definition

An absolute minimum on \((-\infty,\infty)\) is the lowest point of the entire function.

Step2: Analyze the graph

As \(x\to\pm\infty\), the function \(y = w(x)\) seems to approach a horizontal asymptote. The lowest point in the visible part of the graph is \((2,-3)\), but since the function extends to \(-\infty\) and \(\infty\), there is no absolute minimum.

Step1: Recall absolute - maximum definition

An absolute maximum on \((-\infty,\infty)\) is the highest point of the entire function.

Step2: Analyze the graph

As \(x\to\pm\infty\), the function \(y = w(x)\) seems to approach a horizontal asymptote. The highest point in the visible part of the graph is \((0,1)\), but since the function extends to \(-\infty\) and \(\infty\), there is no absolute maximum.

Answer:

A. From the graph, the local minimum occurs at the point \((2,-3)\)

B.