Consider the function f(x) = x/(x-1)

Are there any turning points? Explain how this could help you graph f(x) for large values of x?

Ans: turning points is another word for checking the concavity, and therefore i find the second derivative and equate it to zero and see where it is concave up and down, using that information i can know when the graph is increasing or decreasing, with its end behaviors near the asymptote
I get the second derivative as f''(x) = 2/(x-1)^3. If f''(x) = 0 then there are no solutions, hence there are no turning points.

Am i correct? But how does that help me graph f(x) for large values.

"Turning points" is the same as maximum or minimum points, and we find them by setting the first derivative equal to zero.

Setting the second derivative equal to zero gives us
"points of inflection"

This page summarizes it for you

http://www.teacherschoice.com.au/Maths_Library/Calculus/stationary_points.htm

In most cases you can use common sense to find what happens to the graph for large values of x, by merely considering the numbers involved.

in your case consider a large x like x = 1000
the denominator is simply one less than the top, so we get 1000/999 which is barely above 1
the larger the x the closer we get to 1, but still above it
so for x ---> + infinity, f(x) = 1+
similarly for huge negative x's, e.g. x = -1000
we get -1000/-1001 which is barely below 1
so as x ---> - infinity , f(x) ---> 1-

what is the anwser to this input output table here it is input(x)-2,-1,0,1,2,3,here is the output(y)-1,1,3,5,7,9,

3(x^2-x-6)/4(x^2-9) what is the domain?

Yes, you are correct in finding the second derivative and equating it to zero to determine the turning points or the concavity of the function. However, in the case of the function f(x) = x/(x-1), the second derivative is actually f''(x) = 2/(x-1)^3, not 2/(x-1)^2 as stated earlier.

To understand how this information helps you graph f(x) for large values of x, let's analyze the behavior of the function:

When x approaches positive infinity (large values of x), the function f(x) will approach a horizontal asymptote. In this case, the function approaches y = 1. This means that as x gets larger and larger, f(x) gets closer and closer to the value 1.

By analyzing the concavity, we can determine how f(x) approaches the asymptote. Since the second derivative is always positive (2/(x-1)^3 > 0 for all x), we know that the graph is concave up for all x.

Therefore, as x approaches positive infinity, the function f(x) will be increasing and concave up. This information helps us accurately sketch the graph and predict its behavior as x becomes larger.