Question

. The figure below shows the slope field for a differential equation . Let be the family of functions that are solutions of the differential equati

(a) Determine to the nearest integer the value of x for which all of the members of the family of g(x) will have a relative minimum value. Explain how you know.



(b) Determine to the nearest integer the value of x for which all of the members of the family of g(x) will have a relative maximum value. Explain how you know.



(c) On the figure below

Answer 1

Since you're in differential equations (apparently), you've already had calculus. One of the first things we learn in calculus about applications of derivatives is that when the derivative changes from negative to positive at some x-value, there's a relative minimum there.

Answer 2

So I look carefully at the x-axis in the illustration you've attached, and try to find x values at which the derivative is negative on the left and positive on the right. Can you identify any such x-value or values?

Answer 3

5 is going to be the biggest right because that is as far as the graph goes

Answer 4

Look at x=0. I see that the derivative is negative (below the x-axis) on the left of x=0 and positive (above the x-axis) on the right of x=0. That meets all of the requirements for a relative minimum at x=0.

Answer 5

We're not looking for anything "biggest," but rather are looking for x value(s) at which the derivative (in the diagram) is - on the left, then 0, and then + on the right.

Answer 6

I sincerely apologize, but I've been asked to come to dinner and do not want to keep my hosts waiting. Hopefully another person can help you; if not, I'll try to get back to you later this evening. Good luck!

Answer 7

Thank you!

Answer 8

I'm now in Las Vegas.
I've looked at your diagram again and have realized that the solutions to this problem are easier to find than I'd first thought.
You've surely heard of "connect the dots."
Here, we don't have dots, but you could connect some of the short, little dashes that represent the slope of the tangent line at any point on the given graph.
If you look carefully at x=2, you'll probably see that any curve made up of connected dashes appears to have a local minimum at x = 2 and a local max at x = -1.
I see y ou're offline right now. If you're still interested in discussing this problem, send me a message through OpenStudy; I'll respond as soon as possible (but will be on the road a lot tomorrow also).