`f(x) = (x^2 + 1)/(x^2 - 1)` Determine the open intervals on which the graph is concave upward or downward.

Textbook Question

Chapter 3, 3.4 - Problem 9 - Calculus of a Single Variable (10th Edition, Ron Larson).
See all solutions for this textbook.

1 Answer | Add Yours

gsarora17's profile pic

gsarora17 | (Level 2) Associate Educator

Posted on


differentiating by applying quotient rule,




differentiating again,




In order to determine the concavity , determine when f''(x)=0 , however there are no points at which f''(x)=0 but at x=`+-` 1 the function is not continuous.

So test for concavity in the intervals (-`oo` ,-1) , (-1,1) , (1,`oo` ) by plugging in the test values in f''(x).




Since f''(2) and f''(-2) are positive , so the function is concave upward in the interval (-`oo` ,-1) and (1,`oo` )

f''(0) is negative , so the function is concave downward in the interval (-1,1). 

We’ve answered 319,644 questions. We can answer yours, too.

Ask a question