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

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gsarora17 eNotes educator| Certified Educator


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).