`f(x) = (2x^2)/(3x^2 + 1)` Determine the open intervals on whcih the graph is concave upward or downward.

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gsarora17 | (Level 2) Associate Educator

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differentiating by applying quotient rule,




differentiating again,




In order to determine the concavity , first determine when f''(x)=0,




Now let us consider the intervals (-`oo` ,-1/3) , (-1/3,1/3) and (1/3,`oo` ) and determine the signs of f''(x) by plugging in the test values in f''(x).




So the graph is concave downward in the interval (-`oo` ,-1/3) and (1/3,`oo` )

graph is concave upward in the interval (-1/3,1/3).