The differential equation dy/dx=(y-2)/(y+1)

produces a slope field with horizontal tangents at y = 2
produces a slope field with vertical tangents at y = −1
produces a slope field with columns of parallel segments

a) I only <------- My answer ( This is correct??)
b) II only
c) I y II only
d) III only

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Since dy/dx = (y-2)/(y+1)
you are correct that I is correct.
But, II must also be correct, since you are trying to divide by zero

Since dy/dx is independent of x, it must produce rows of parallel segments.
But surely not columns, since y varies as you move vertically.

So, C is the answer.

I am between answer A and C???

Can you help me please????

Thank you

To determine which options are correct, we need to analyze the given differential equation and understand the implications it has on the slope field.

The differential equation given is dy/dx = (y - 2)/(y + 1).

Now let's analyze each option:

I. Produces a slope field with horizontal tangents at y = 2

To determine if this is true, we need to find the critical points of the differential equation. Critical points occur when the numerator of the differential equation is equal to zero.

Setting (y - 2) = 0, we find y = 2 as a critical point.

Now, let's evaluate the sign of dy/dx for values of y less than and greater than 2:

For y < 2, (y - 2) is negative, and (y + 1) is positive. Thus, dy/dx is negative.

For y > 2, (y - 2) is positive, and (y + 1) is positive. Thus, dy/dx is positive.

From this analysis, we can conclude that dy/dx changes from negative to positive as we move from left to right through the critical point y = 2. This means that the slope field has horizontal tangents at y = 2. Therefore, option I is correct.

II. Produces a slope field with vertical tangents at y = -1

To determine if this is true, we need to check if there are any critical points at y = -1.

Setting (y + 1) = 0, we find y = -1 as a critical point.

Now, let's evaluate the sign of dy/dx for values of y less than and greater than -1:

For y < -1, (y - 2) is negative, and (y + 1) is negative. Thus, dy/dx is positive.

For y > -1, (y - 2) is negative, and (y + 1) is positive. Thus, dy/dx is negative.

From this analysis, we can conclude that dy/dx changes from positive to negative as we move from top to bottom through the critical point y = -1. This means that the slope field has vertical tangents at y = -1. Therefore, option II is correct.

III. Produces a slope field with columns of parallel segments

To determine if this is true, we need to observe the behavior of dy/dx across the entire range of y-values.

From our previous analysis, we found that dy/dx changes from negative to positive at y = 2 and from positive to negative at y = -1. This means that the slope field will have segments that change in direction at these critical points. However, these segments will not be parallel columns.

Therefore, option III is not correct.

Based on this analysis, the correct answer is a) I only, as both options I and II are correct.