without using calculus?

Question

show the function y = x^2 / (x+1)(x-2)

never has any values between 0 and 8/9.

I can show that when y = 0 the function has the root of x = 0 twice. Implying a local maximum or minimum. Then by taking x = 0.1 and -0.1 I can show it is a maximum.

Then taking y = 8/9 the root is x = -4 twice. Then by taking
x = -4.1 & -3.9, this is a minimum. Does this prove the above question, do I need more, or do I need to do it in a completely different method?

Does the fact that the function is quadratic mean that the fuction will only hit these points twice (even though the twice in this case is at the same x-value) even if the domain is all the real numbers?

If so this means that the function can never pass into these points again. Right?

Answer 1

"I can show that when y = 0 the function has the root of x = 0 twice"
That is clear, y=0 implies x=0. But I would say that it has one root, not two equal roots.
" Implying a local maximum or minimum"
no, the local maximum and minimums are not necessarily in the points where you find a root.
"Then by taking x = 0.1 and -0.1 I can show it is a maximum."
I don't see how can you show that, in 0.1 and -0.1 the function is still negative, if that would count for anything.

"Does this prove the above question, do I need more, or do I need to do it in a completely different method?"
It doesn't prove anything. Yes, you need a different method.
You take the derivative, find the monotony intervals. If in doubt, you can make the graph of the function.


"Does the fact that the function is quadratic mean that the fuction will only hit these points twice"
The function is not quadratic. It is a rational function with quadratic numerator and denumerator.

"If so this means that the function can never pass into these points again. Right?"
Well, yes. They can't reach 8/9 and 0 in other points but how exactly shows this that y hasn't any values between 0 and 8/9?

I appreciate though that you try to find an easy way out.

Answer 2

Well, 1/2 = .5 lies between 0 and 8/9.
So y = .5^2/[(.5+1)(.5 - 1.5)]
=.25/[(1.5)(-1.5)]
= .25/-2.25 = 1/9
So your assumption is wrong.

If you meant y = x^2(x-2)/(x+1), then
y = .25(-1.5)/1.5 = -.25 so again your assumption is wrong.

Suggest you make sure you have stated the problem correctly.