4.3 max & min on an interval\n8. a. find the (x,y) coordinates of all local minimums of w(x).\nb. find the…

4.3 max & min on an interval\n8. a. find the (x,y) coordinates of all local minimums of w(x).\nb. find the (x,y) coordinates of all local maximums of w(x).\nc. does w(x) have an absolute minimum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this minimum.\nd. does w(x) have an absolute maximum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this maximum.\ne. find the (x,y) coordinates of the absolute maximum(s) of (x,y) on -1,1.\nf. find the (x,y) coordinates of the absolute minimum(s) of (x,y) on -1,1.

4.3 max & min on an interval\n8. a. find the (x,y) coordinates of all local minimums of w(x).\nb. find the (x,y) coordinates of all local maximums of w(x).\nc. does w(x) have an absolute minimum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this minimum.\nd. does w(x) have an absolute maximum on (-∞,∞)? if so, give the (x,y) coordinates for each occurrence of this maximum.\ne. find the (x,y) coordinates of the absolute maximum(s) of (x,y) on -1,1.\nf. find the (x,y) coordinates of the absolute minimum(s) of (x,y) on -1,1.

Answer

Explanation:

Step1: Recall local - minimum definition

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

Step2: Identify local minima from the graph

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

Answer:

A. The local minimum occurs at approximately ((2,-3))

Step3: Recall local - maximum definition

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

Step4: Identify local maxima from the graph

From the graph, we look for the points where the curve peaks.

Answer:

B. The local maximum occurs at approximately ((0,1))

Step5: Recall absolute - minimum definition

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

Step6: Analyze the graph for absolute minimum

As (x\to\pm\infty), (y = w(x)\to-\infty), so there is no absolute minimum on ((-\infty,\infty))

Answer:

C. No

Step7: Recall absolute - maximum definition

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

Step8: Analyze the graph for absolute maximum

The graph does not have a highest point as it is not bounded above in the domain ((-\infty,\infty))

Answer:

D. No

Step9: Consider the interval ([-1,1])

We only look at the part of the graph where (-1\leq x\leq1)

Step10: Identify the highest and lowest points in ([-1,1])

The highest point in the interval ([-1,1]) is ((0,1)) and the lowest point is ((-1,-1))

Answer:

E. The absolute maximum on ([-1,1]) is ((0,1)) F. The absolute minimum on ([-1,1]) is ((-1,-1))