Consider the differential equation

Let be a solution to the equation with the initial condition . Without attempting to solve this equation explicitly answer the following questions.

- Since we also have . Does have a relative maximum, relative minimum, or neither at 0?
- If then and if then . Find positive numbers and such that
- Prove that
- Prove that
for some finite number and find the value of .

**Incomplete.**

a.) From Theorem 4.9 on page 189, we can determine if a local extreme is a maximum, minimum, or neither. So, (assuming the initial conditions are met), if we take the second derivative of y, at x = 0, with y(0) = 0 we get

Since this value is greater than 0, from Theorem 4.9 we can conclude that this extreme is a local minimum.

b.) We know that for x >= 10/3, f'(x) >= 2/3. And we know that for x>0, y>0. So, we can use the point-slope form of a line to exhibit a and b such that

Let

Then

Or

c.) We can use the above values for a and b to show that this is the case. Since we know that for all x >= 10/3

We also know that for all x >= 10/3

Now, if we take the following limit

We can see that the limit goes to 0

But we know that

So if the above limit goes to zero as x goes to infinity, then

Goes to zero as well.

d.) We can once again use the results from part (b.). Since we know that for all x >= 10/3

We can use little-o notation from section 7.9 to show that that

I think I can clarify part (c.) by using o-notation there as well.

So, if we have the limit

We can use o-notation to write the above limit as

And we can use Theorem 7.8 (c.) to simplify terms, giving us

As x goes to plus infinity, the limit goes to zero.

I don’t think you can start with the assumption that y = 2/3x + o(1).

To me it points to the L’Hospital \infty/\infty direction, even though we didn’t prove it in the book until this point.