The discussion focuses on analyzing the function f(x) = x²/(x-1) to identify asymptotes, extrema, and points of inflection. The first derivative is correctly derived as f'(x) = x(x-2)/(x-1)². The second derivative indicates a change in concavity at x = 1, confirming that there is indeed a point of inflection at this value. The use of the quotient rule and chain rule is emphasized for accurate differentiation.
PREREQUISITESStudents studying calculus, particularly those focusing on function analysis, derivatives, and critical points. This discussion is beneficial for anyone seeking to improve their understanding of differentiation techniques and their applications in identifying function behavior.
Write this as an equation, and use parentheses.mathpat said:Homework Statement
x^2 / x-1. Identify any asymptotes, extrema and points of inflection.
As an equation, this is f'(x) = x(x-2)/(x-1)2.mathpat said:Homework Equations
The Attempt at a Solution
I am stuck trying to derive my first derivative. My first derivative equals x(x-2)/(x-1)^2.
Your answer is correct, and this can be verified by calculating the derivative using the technique that Zondrina suggested.mathpat said:I tried to use the quotient rule again using while incorporating the chain rule and after multiple attempts I kept resulting with 2 / (x-1)^3.
Any suggestions?
The sign of f'' changes at x = 1. If x < 1, f''(x) < 0, and if x > 1, f''(x) > 0. Does this mean that there is an inflection point at x = 1? Why or why not?mathpat said:I see... so basically that means there is no point of inflection since I can not set that equation equal to zero and solve?
mathpat said:By the way, I appreciate both of your help.