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Concavity (Posted on 2014-08-26) Difficulty: 3 of 5
For the function below, determine the range of x-values for which the function is concave up:

1) Assuming k is negative.
2) Assuming k is positive.

No Solution Yet Submitted by Dustin    
Rating: 4.0000 (1 votes)

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Solution Solution Comment 2 of 2 |
y = - (x2 + 1/k)-1

Differentiating:      y      = 2x(x2 + 1/k)-2

and again:             y     = 2(x2 + 1/k)-2 4x(x2 + 1/k)-32x

                                    = 2(x2 + 1/k)-3(x2 + 1/k 4x2)

                                    = 2(x2 + 1/k)-3(1/k 3x2)

(1)  For k < 0: 1/k 3x2 < 0,  so the function will be concave upwards

                        (i.e. y > 0) iff  x2 + 1/k < 0, giving |x| < sqrt(-1/k).

            i.e. between the asymptotes, x = +/-sqrt(-1/k), as Charlie found,

            with a local minimum at (0, -k), above the origin.

(2)  For k>0:
   (x2 + 1/k)-3 > 0, so the function will be concave upwards

                        where   1/k 3x2 > 0, giving  |x| < 1/sqrt(3k).

            i.e. between the points of inflection at (+/-1/sqrt(3k), -3k/4),

            with a local minimum at (0, -k), below the origin.




  Posted by Harry on 2014-08-27 13:00:37
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