`y = 2x - tan(x) , (-pi/2, pi/2)` Determine the open intervals on whcih the graph is concave upward or downward.

Textbook Question

Chapter 3, 3.4 - Problem 13 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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differentiating again,





In order to determine the concavity , determine the value of x when y''=0


sinx=0 , so x=0 , pi, 2pi ,.....

Now let us test for concavity in the intervals (-pi/2,0) and (0,pi/2)

y''(-pi/4)=2*(-pi/4)- tan(-pi/4)= -pi/2+1 ( negative)

y''(pi/4)=2*(pi/4)-tan(pi/4)=pi/2-1 (positive)

So, the graph is concave upward in the interval (0,pi/2) and

concave downward in the interval (-pi/2,0)



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