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
Correct.
Did you find any turning points?
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.
To determine if there are any turning points in the function f(x) = x/(x-1), we need to find the second derivative and check where it equals zero. The second derivative tells us about the concavity of the function.
First, we find the first derivative of f(x) with respect to x:
f'(x) = (1*(x-1) - x*(1))/(x-1)^2
Simplifying this expression, we get:
f'(x) = -1/(x-1)^2
Now, we find the second derivative by taking the derivative of f'(x):
f''(x) = (-1*2(x-1))/((x-1)^2)^2
Simplifying further, we get:
f''(x) = -2/(x-1)^3
To check if there are any turning points, we equate f''(x) to zero:
-2/(x-1)^3 = 0
Since the denominator cannot be zero, there are no values of x for which f''(x) equals zero. Therefore, there are no turning points in the function f(x) = x/(x-1).
Now, to graph the function for large values of x, we can consider its end behavior and asymptotes. The function has a vertical asymptote at x = 1 because the denominator (x-1) approaches zero as x approaches 1. As x becomes larger, the function approaches 0 since the numerator, x, increases at a slower rate than the denominator, (x-1). Therefore, the graph approaches the x-axis as x gets larger.
Overall, the lack of turning points in the function f(x) = x/(x-1) indicates that the function does not have any local extrema. By considering the end behavior and asymptotes, we can sketch the graph for large values of x.